26: Chapter 26 The Foundation of Concepts

[For any ε > 0, there exists a positive integer N such that when n > N, we have |aₙ - A| < ε.]

Jiang Lin was still staring at this line of words on the screen.

On a draft notebook spread out beside the computer,

four words were written at the very top.

[Advanced Mathematics]

Below it was a line of smaller text.

[First Rule: If you don't understand the definition, you are not allowed to solve problems.]

He had written this line of text last night.

He was full of imposing momentum when writing it.

When it actually came to execution, Jiang Lin discovered that the so-called "no solving problems without understanding definitions" sounded like an inspirational maxim, but in actual operation, it felt like taking a habit, hacking it open with a knife, and sewing it back together in another way.

During high school, he was used to problems giving conditions and asking for conclusions.

If he couldn't do it, he looked at the example problems; if he couldn't understand the example problems, he looked at the answers.

If he couldn't understand the answers, he would copy them once; if he still didn't understand after copying once, he would break down the steps.

Breaking them down to the end, he could always find a few familiar things, such as formulas, models, images, or substitution...

During those nine years in the third Wasteland, he relied on this method to stubbornly grind himself from seventy-first in his grade to first in the entire city.

It was the same for mathematics, Physics, and also for chemistry and biology.

But college mathematics was not like this.

College mathematics didn't give problems right off the bat; it gave you a concept first.

And then it required you to believe that an entire forest would grow behind this concept.

Limits, continuity, derivatives, integrals, series...

Placed in the table of contents, these words looked like old acquaintances.

He had learned derivatives in high school, and he had seen a bit of integrals too.

Not to mention vectors, which were used every day in Gaokao analytic geometry.

But when he clicked in and took a look, Jiang Lin discovered that those familiar words in high school mathematics were merely the signboard at the door.

The real house was in the back, and it was frighteningly dark inside.

He looked back at the screen.

In the instructional video, the college professor's voice was unhurried.

"For the strict definition of the limit of a sequence, grasp two quantities: one is ε, and the other is N. ε represents the error you allow, and N represents starting from which term onward..."

The professor explained it very clearly.

At least the speaking speed was clear, the articulation was clear, and the blackboard writing was also clear.

But as Jiang Lin listened, his brain still began to slip away.

It wasn't getting distracted; it was that he couldn't grasp it.

ε can be arbitrarily small.

What does arbitrarily small mean?

It can be as small as 0.1, as small as 0.01, or even as small as 0.000001.

As long as you give a positive number, I can always find an N so that all subsequent terms enter this error range.

He could repeat this sentence, and even recite it from memory.

But memorizing it was useless; he didn't know if he truly understood it.

It was like a person who couldn't swim standing on the bank and memorizing the breaststroke movements inside out, but once they actually got into the water, they would still thrash about twice and sink straight down.

Jiang Lin paused the video at the frame where the professor finished writing the definition.

He picked up a ballpoint pen and rewrote the definition in the draft notebook.

[∀ ε > 0, ∃ N ∈ N⁺, when n > N, |aₙ - A| < ε.]

After writing it, he stared at it.

A minute later, he put down his pen, stood up, and walked out of the Stone House.

The wind immediately blew against his face.

Twelve days had already passed in the fourth Wasteland, and the weather was a bit warmer than when he left during the third time.

The seasons of the Wasteland were not completely identical to Earth, but after recording for the previous ten years, he had roughly figured out some patterns.

This Wasteland planet also had hot and cold cycles, except that the period was slightly longer than Earth's, the diurnal temperature range was greater, the wind was harsher, and the rain was more acidic.

To the south of the Stone House, that two-hundred-square-meter plot of land had its surface soil color darkened significantly after nine years of plowing, spreading ash, green manure, composting, and repeated backfilling of Pumpkin vines and Soybean stalks.

It wasn't the oily, moist black of normal land, but rather resembled dark red mixed with grayish-brown, like a piece of old cloth that had been rubbed repeatedly.

Fragmented plant residue began to appear inside.

Moss had been transplanted bit by bit by him from beneath the riverbed's rock eaves, and now several grayish-green patches had grown on the edges of the field ridges.

Like a layer of mold stuck to a stone.

But on the Wasteland, this was a miracle.

Jiang Lin walked to the edge of the field and squatted down to look at a Soybean seedling.

The Soybean seedling had grown to the height of a calf, its leaves were darker in color than Soybeans on Earth, its leaf surface was rough, and the edges had a slight curl.

During the nine years of the third Wasteland, through seed saving and selection, he had already made these Soybeans adapt to the acidic soil, weak light, and low temperature of the Wasteland.

Of course, saying they adapted was a bit of an exaggeration.

To be exact, most of them died, and the ones left behind could barely survive.

Jiang Lin reached out and gently brushed aside the soil near the roots of the Soybean.

The root system pierced downwards densely and finely, and a few light brown small nodules hung beside the taproot.

Root nodules.

When he first saw this thing in the Wasteland, he was so excited he almost pulled out the entire Soybean plant.

This meant that rhizobia could also survive in the Wasteland environment.

It meant that the nitrogen in the air could be fixed into the soil bit by bit by these insignificant little lives.

It meant that this patch of land was not completely dead.

Jiang Lin squatted by the edge of the field, looking at those few small root nodules, and suddenly remembered that line of definition from just now.

For any ε.

There exists N.

When n is greater than N.

All enter the error range.

He narrowed his eyes.

What if this field was treated as a sequence?

In the first year, the soil pH was unstable, and crop yields fluctuated greatly.

In the second year, the yield was still very low, but slightly better than the first year.

The third year, the fourth year, the fifth year...

Every year, he recorded the potato yield per square meter, Soybean plant height, Pumpkin fruit-setting quantity, rainwater pH, soil acidity/alkalinity, and compost decomposition cycle.

Those data points did not rise smoothly.

One year there was too much acid rain, and half the potatoes rotted.

One year a storm snapped the Pumpkin vines.

Another year he spread too much plant ash, causing the leaves of several ridges of Soybeans to turn yellow, almost burning the seedlings.

But from a longer scale, the yield was indeed approaching a certain stable value.

At the beginning, a square meter of potatoes could only harvest a little over a catty, later two catties, and then three catties.

By the late stage of the third Wasteland, in some plots with better maturation, one square meter could yield close to four catties.

It did not grow infinitely; it was merely approaching an upper limit.

This upper limit might be the ultimate yield jointly determined by the Wasteland's sunlight, water source, soil, and crop varieties.

If a permitted error is given.

For example, the error of the potato yield per square meter from 3.8 catties is less than 0.5 catties.

Then after which year can most of the subsequent yields fall within this range?

What if the error is required to be less than 0.2 catties?

After which year should it be?

What if the error is required to be as small as 0.1 catties?

Jiang Lin slowly stood up.

The wind blew across the field ridges, and the Soybean leaves made a subtle frictional sound.

He suddenly turned around, returned to the Stone House, sat in front of the desk, and picked up the pen again.

This time, he didn't copy the definition directly.

[Example: Wasteland Potato Field Yield Sequence.]

First year: 1.3 catties/square meter.

Second year: 1.8 catties/square meter.

Third year: 2.2 catties/square meter.

Fourth year: 2.7 catties/square meter.

Fifth year: 3.1 catties/square meter.

Sixth year: 3.3 catties/square meter.

Seventh year: 3.5 catties/square meter.

Eighth year: 3.6 catties/square meter.

Ninth year: 3.72 catties/square meter.

Of course, the real data wasn't this clean.

He just wrote an idealized model first for the sake of understanding.

Then he wrote beside it.

[A = 3.8]

[When ε = 0.5, starting from the fifth year, all yields are between 3.3 and 4.3.]

[When ε = 0.2, starting from the eighth year, all yields are between 3.6 and 4.0.]

[When ε = 0.1, starting from the ninth year, all yields are between 3.7 and 3.9.]

He paused for a moment, then wrote below.

[The smaller ε is, the stricter the requirement, and the larger N may be needed.]

After writing this sentence, his spine tingled a little.

It wasn't because he finally understood completely.

It was just that a crack in the door had opened a bit.

That line of definition, which originally felt like a spell, suddenly gained a little warmth.

It was no longer just about for any ε there exists N.

It was as if it were saying, you can limit the error to be as small as you want.

As long as this sequence is truly approaching A, then sooner or later, it will steadily enter the range you specified and never run out again.

Jiang Lin wrote down this explanation.

[Limits are not about whether the front is chaotic.]

[Limits are about whether the back can stabilize.]

[The front can be chaotic, fluctuating, mistaken, and failing. As long as from a certain position onward, it entirely approaches the target, it is called convergence.]

The pen tip stopped on the paper surface.

Jiang Lin stared at the two words "convergence", and suddenly smiled.

He felt this word suited him quite well.

For the first decade and more before his third year of high school, he had always been chaotic—his grades were chaotic, his mindset was chaotic, and his future was chaotic.

In the first Wasteland, he couldn't even get water to drink.

In the second Wasteland, he relied on compressed biscuits and polluted water to last over forty days.

In the third Wasteland, plowing the land until his palms blistered and his legs cramped, he didn't know how many times he had failed over nine years.

But if his entire life was also viewed as a sequence,

then starting from a certain N, was he finally beginning to converge toward a certain target?

Jiang Lin did not continue thinking about it.

He lowered his head and wrote today's date at the very top of the draft notebook.

[Fourth Wasteland, Day 12.]

[Task: Understand the definition of the limit of a sequence.]

[Method: Do not memorize the definition; find Real World counterparts first.]

After writing, he played the video again.

The professor went from the limit of a sequence to the limit of a function.

[When x approaches x₀, f(x) approaches A.]

Then he paused again and wrote a page.

[x approaches, it does not mean x actually arrives.]

[Like me walking from the Stone House toward the puddle. The distance gets smaller and smaller, but I don't have to stand at the center of the puddle.]

[Function values approaching is where positional changes cause result changes.]

He drew a crooked line.

On the left, he labeled it Stone House, on the right, he labeled it puddle, and in the middle, he drew a few points.

Write x₁, x₂ and x₃ below.

The closer you get to the puddle, the higher the air humidity, the damper the ground, and the heavier the mineral sourness.

If you take the position from the center of the puddle as x, and the concentration of that sourness in the air as f(x), then the closer a person gets to the puddle, the closer f(x) gets to a certain value.

But you don't need to actually jump into the puddle; you are just getting closer.

Jiang Lin knew this example was very crude and not rigorous.

But it was useful.

Being useful was enough.

He was never the kind of genius who could see through to the essence at a glance.

His learning method had always been clumsy, too.

He just tried to drag abstract things onto the ground, push them into the mud, and knead them into his own life.

Only in this way would it stop floating around.

In the morning, he studied four pages, four pages of Advanced Mathematics.

Two of which were still the table of contents, introduction, and list of symbols.

But he wasn't as anxious as before.

Because this time, what remained on the draft paper was not copied definitions, but a bunch of clumsy yet uniquely his own explanations.

At noon, he cooked a pot of potato and Soybean paste.

Calling it cooking was actually just using a small stove built with Wasteland stones, burning dead wood and dry Pumpkin vines, and simmering chopped potatoes, soaked Soybeans, and a bit of fragmented Moss in a metal lunch box.

The water was treated acid rain water.

First filtered with activated carbon and cotton, then adjusted to near neutral with baking soda, and finally boiled.

The taste was terrible.

The potatoes had a faint mineral fishy smell, the Soybeans didn't smell fragrant enough when cooked due to variety and environmental reasons, and the Moss carried a lingering bitterness.

He sprinkled a little salt into it.

That bit of salt was brought from the Real World, extremely precious.

Jiang Lin held the lunch box, sat at the entrance of the Stone House, and ate it bite by bite.

As he ate, he thought of The Feynman Lectures on Physics downloaded on his computer.

He originally wanted to make a start on Advanced Mathematics within a month before starting General Physics.

Now it seemed that it would be good enough if he could chew through and understand limits and continuity within a month.

Even limits and continuity might not be chewed through and understood.

Jiang Lin finished the last bite of the potato paste, rinsed the lunch box with clean water, and drank the starchy water as well.

He returned to the desk and didn't continue reading further.

Instead, he opened a new section on his draft notebook.

[Concept Notebook]

The first page read: [Limits]

Below it was divided into five columns.

First column: Textbook definition.

Second column: Plain human explanation.

Third column: Real World example.

Fourth column: Counterexample.

Fifth column: Common question types.

He first copied the definition of the limit of a sequence into the first column.

Then he wrote in the plain human explanation: [No matter how close you demand to be to the target, as long as you go far enough down the line, everything can enter this range.]

Real World example wrote: [Wasteland potato field yields stabilize year by year.]

He thought for a long time in the counterexample column.

Writing: [If a sequence approaches A for a while and then runs far away, and it's always like this, there is no limit.]

He gave the simplest one: [aₙ = (-1)ⁿ]

Sometimes it's 1, sometimes it's -1.

It jumps forever and doesn't converge.

He wrote beside it: [Like Sun Ming saying he would work hard the day before the exam, and playing games again the next day. The state jumps between 1 and -1, having no limit.]

After writing, he couldn't help laughing out loud by itself.

There was no Sun Ming in the Stone House, only the sound of the wind squeezing through the cracks of the door.

After Jiang Lin finished laughing, he felt a bit of longing for that big-headed fatty.

Missing him chewing spicy strips while asking: "Old Jiang, why does this question choose C?"

Missing him clutching his head in pain after listening to the explanation: "Damn, I shouldn't have grown two hands."

Of course, this longing was short, so short that it was like wind blowing across a lamp wick, passing by in a flash.

Jiang Lin lowered his head and continued writing.

In the afternoon, he started looking at example problems.

First example problem: Prove that the limit of the sequence 1/n is 0.

This problem, viewed with high school thinking, was too simple.

The larger n is, the smaller 1/n is, of course approaching 0.

Is proof even needed?

But university mathematics required you to prove it, proving it with definitions.

Jiang Lin initially thought it was superfluous.

Later, biting the bullet and reading on, he understood that this was not to prove that 1/n would approach 0.

This was to train you to use definitions.

Problem requirement: For any ε > 0, find N such that when n > N, |1/n-0| < ε.

Which is 1/n < ε.

Equivalent to n > 1/ε.

Therefore, it is sufficient to take N > 1/ε.

The answer was very short.

So short that it made people uncomfortable.

Jiang Lin copied it once.

Didn't fully understand it.

Copied it again.

Still felt something was wrong.

He stopped and gave specific numerical values on the draft paper.

If ε = 0.1, then as long as n > 10, 1/n is less than 0.1.

If ε = 0.01, then as long as n > 100, 1/n is less than 0.01.

If ε = 0.000001, then as long as n > 1000000, 1/n is less than 0.000001.

Therefore N changes with ε.

ε is not fixed.

N is not fixed either.

However strict the requirements others give you, you find a starting point far enough back.

This is the core of proof.

Jiang Lin wrote beside it: [Proving a limit means accepting any harassment from the other party, and then providing a starting point far enough back.]

After writing, he felt this sentence was very much like the Wasteland.

The Wasteland was also arbitrarily harassing him.

Acid rain, heavy metals, storms, loneliness...

It gives an ε.

He finds an N.

If the first year doesn't work, then the second year.

If the second year doesn't work, then the fifth year.

If five years don't work, then ten years.

As long as he wasn't dead, as long as he could keep moving forward, there would always be a position that would turn those once unbearable things into controllable errors.

In the evening, the sun sank near the horizon.

Dark red light slanted in from the window of the Stone House, shining on the draft notebook.

Jiang Lin glanced at today's achievements.

Advanced Mathematics textbook, progressed by four pages.

Video, progressed by twenty-seven minutes.

Draft paper, nineteen pages written.

Concept notebook, completed the initial version of sequence limits.

Example problems, truly understood two.

Jiang Lin didn't curse himself.

He closed those two example problems again, covered the answers, and independently wrote out the proof on a blank sheet of paper.

The first one, 1/n approaches 0.

The second one, (2n+1)/n approaches 2.

The second one was slightly more complex.

|(2n+1)/n-2| = |1/n| = 1/n.

Same as the first one.

Take N > 1/ε.

Proof complete.

Not a single quantifier was omitted.

For any ε > 0.

Take a positive integer N such that N > 1/ε.

When n > N, there is...

Writing to the last line.

Therefore lim (2n+1)/n = 2.

Jiang Lin put down his pen.

Outside the house, the wind was howling; inside the house, the light was steady.

He suddenly realized that this might be the very first brick of his university studies.

Not Quantum Mechanics, not General Physics.

Not those universes, particles, black holes, time, and space that sounded like they could make one's blood boil.

Instead, it was a proof problem so simple it was almost shabby.

Prove that 1/n approaches 0.

But if this brick wasn't laid steadily, everything behind it would be crooked.

Jiang Lin opened the notebook and wrote at the end of today's record.

[Day 12.]

[Advanced Mathematics progress is extremely slow.]

[But today I understood for the first time: university is not about more problems, but deeper definitions.]

[From now on, every time I learn a new concept, I must complete five columns: definition, plain language, Real World example, counterexample, question types.]

[Not allowed to lie to yourself.]

The last four words penetrated the back of the paper.

Because doing problems can be an act, watching videos can be an act, and copying notes can be an act even more.

Filling a notebook of draft paper makes you look hardworking.

Studying twelve hours a day makes you look diligent.

But whether you truly understand or not, only you know.

Jiang Lin used to be best at lying to himself.

Saying things like I almost know how to do this problem, saying things like I'll make up for this knowledge point later, saying things like this time was just carelessness, saying things like as long as I work a bit harder, my grades will go up...

It was still the Wasteland that taught him reality wouldn't let you off the hook because of your self-consolation.

No water means no water, no food means no food.

If the soil pH value is wrong, the seeds will rot.

Saying you've worked hard is useless; you have to actually treat the water until it's drinkable.

Actually loosen the soil, actually let the Soybeans grow root nodules.

Learning is the same.

Saying you understand is useless.

You have to be able to write the proof without answers, be able to explain it clearly to Sun Ming, and be able to explain it clearly to yourself from ten years ago.

The night completely pressed down.

Jiang Lin closed the computer, pressed the draft papers filled today in order under stones to prevent them from being blown away by the wind.

He checked the battery power again.

The solar panel charged well today, with six hours of effective sunlight during the day, and the battery could still support the computer for two hours of use and the camping light until midnight.

But he didn't stay up late anymore.

University courses are not a high school sprint; stubbornly pushing through by staying up late is meaningless.

Especially in the Wasteland, the body is the only capital.

He walked out of the Stone House and inspected the fields one last time.

Potato leaves swayed slightly in the night wind.

Pumpkin vines sprawled beside the ridge, like strips of silent spicy strips.

Soybean seedlings stood neatly.

The distant dried-up riverbed left only a deeper shadow in the darkness.

Jiang Lin stood by the field and suddenly said a sentence in English in a low voice.

"I am converging."

After finishing, he paused and corrected himself.

"No."

"I am trying to converge."