Chapter 93
Office hours.
With a knock-knock-, Eun-chae unni’s face peeked through the crack in the door.
“Professor Yoo A-yeon~!”
“Ah, Eun-chae unni!”
Scamper-scamper-scamper-scamper-!
Overjoyed, I sprang up from my child-sized chair, ran over to Eun-chae unni, and threw myself into her arms.
“Nihihi, unni! You really came!”
I buried my face in Eun-chae unni’s arms.
“Eek, Professor! Why are you so cute?!”
Eun-chae unni vigorously patted and squished me.
I may be a dignified professor in the classroom, but in front of her, I turn into nothing more than an adorable little sister who gets showered with affection.
“Hi, Eun-chae.”
“Oh, Professor’s mom! Hello!”
Eun-chae unni also bowed politely to Mom.
Since they often met at the chess club, she and Mom were quite close too.
Once the squishing session finally ended,
Eun-chae unni pulled a Pepero chocolate out of the pocket over her chest.
“Here, one bite for our adorable professor. Open wide-.”
“Aaah-.”
Like a baby bird waiting for its mother to feed it, I opened my mouth wide toward the sky.
Pop-!
The sweet and slightly bitter fragrance of chocolate gently spread throughout my mouth.
“Hwaaah, yummy.”
Chew-chew-!
I smiled with my eyes and put on the happiest expression in the world.
✒️✒️✒️
After the sweet time passed.
Eun-chae unni opened her notebook and brought up the real reason she had come during office hours.
“Professor, about that assignment. I know that 0.9999… equals 1, but when I actually try to prove it using what we learned today, it’s so difficult. I don’t even know where to start.”
Eun-chae unni made a teary, dejected face.
Apparently, she had run into the wailing wall that mathematics newbies were said to encounter.
Epsilon-delta wasn’t an easy concept.
“Unni. Then let me ask you something first.”
I began speaking with a serious expression.
“How do you know that 0.9999… is 1?”
Pop-!
With a strained heave, I pulled the cap off a board marker and handed it to Eun-chae unni.
“Try proving it!”
“Hmm, I know this one!”
✒️✒️✒️
Eun-chae unni confidently wrote an equation on the whiteboard.
It was the familiar proof method she had learned at school.
………………………………
x = 0.999…
Multiplying both sides by 10,
10x = 9.999…
Subtracting the two equations,
10x − x = 9
9x = 9
∴ x = 1
………………………………
“If you do this, x is 1, right?”
Eun-chae unni looked at me with a proud expression.
“Oh, right! There’s this one too.”
………………………………
1/3 = 0.333…
Multiplying both sides by 3,
1 = 0.999…
………………………………
“Like this! What do you think, Professor?”
Eun-chae unni looked at me with sparkling eyes.
‘Hmm… Just as I thought.’
I nodded inwardly.
This method sacrificed logical rigor in favor of being easy to accept at a glance.
If you’re trying to convince a non-major that “0.9999… is 1,” this much is perfectly sufficient.
But!
I, Yoo A-yeon, am a bona fide mathematics professor.
And Eun-chae unni here is a full-fledged undergraduate mathematics major.
I can’t just give her some sloppy, hand-wavy proof like this!
I need to completely overhaul Eun-chae unni’s non-major mindset, one piece at a time!
“Eun-chae unni. But that proof actually contains a logical leap.”
“What?!”
Eun-chae unni jumped in surprise.
“But this is the proof they teach at school…?”
“That’s right. But not everything taught in school is logically sound.”
I climbed onto the footstool and tapped her equations with the board marker.
“Huh…? Why is this wrong…?”
Eun-chae unni tilted her head in confusion.
“Look carefully. Before proving what 0.999… actually is, you already assumed that 0.999… is a proper real number and that you can perform the four arithmetic operations on it.”
I pointed to Eun-chae unni’s first line.
“You set x = 0.999…, then multiplied and subtracted, right? But before you can do that, you have to establish that this is a number that truly exists and that multiplication and subtraction are possible.”
“Uh…”
“The same goes for the one below. The fact that 1/3 = 0.333… has to be proven first. And you have to establish that multiplying this number by 3 is allowed.”
“B-but….”
Eun-chae unni fidgeted awkwardly.
“It obviously looks like something you can perform arithmetic on, though, right? So can’t we just multiply and subtract?”
“You can think of it that way. But, unni, you realize that’s intuition, not logic, right?”
“Ah… Right.”
Eun-chae unni’s expression fell.
I tapped the whiteboard.
“Actually, this is incredibly important. Let me give you a fun example.”
I adjusted my grip on the board marker and wrote a new number on the whiteboard.
“Let’s imagine a number with 9s extending infinitely to the left.”
[…99999]
“…Huh?”
Eun-chae unni looked at it with an expression of doubt, as if wondering, ‘Does a number like that even exist in the first place?’
“I don’t know what this number is supposed to be, but let’s add, subtract, and multiply it for now. Just like you did earlier, Eun-chae unni.”
I quickly wrote out the equations.
………………………………
x = …99999
Multiplying both sides by 10,
10x = …99990
Subtracting the two equations,
9x = −9
∴ x = −1
………………………………
“And ta-da! I never expected it, but goodness! This number with infinitely many 9s to the left turns out to be −1!”
“……Huh?”
Eun-chae unni’s expression went blank.
“H-how can that make sense? There are infinitely many 9s, so how could it be negative…? No, that diverges to infinity in the first place!”
“Right? Just now, you thought, ‘There’s no way such a number exists’ and ‘There’s no way it converges,’ didn’t you?”
“Y-yes!”
“But, unni.”
I gazed steadily into Eun-chae unni’s eyes.
“I want to say the exact same thing to you.”
“……!”
Eun-chae unni’s pupils widened in shock.
“You don’t know whether 0.9999… converges or not. You don’t even know whether a number like this really exists. But you simply assumed that it obviously converges and that you could perform arithmetic on it, then went ahead and added, subtracted, and multiplied, didn’t you?”
I grinned.
When pointing out an undergraduate’s mistake, I become the ultimate bratty Yoo A-yeon!
I just don’t call her a total noob~ noob~.
“So, to avoid making this kind of error, before performing arithmetic, you have to start by defining what kind of number this is.”
“Y-yes. I suppose we do.”
I pointed at the 0.999… on the whiteboard.
“0.999… is exactly the same when you’re first learning about it.”
…
“How it’s defined, whether it converges to a real number, and whether arithmetic can be performed on it—all of that has to be proven first. Only then can you do arithmetic with it! At our current stage, it’s too soon!”
At that moment.
It felt as if something inside Eun-chae unni’s head had shattered with a crash-clatter-.
“W-wait a minute….”
Eun-chae unni clutched her head with both hands.
“Then… was the proof we learned at school entirely… wrong?”
“Strictly speaking, it’s not wrong so much as incomplete. The conclusion is correct, but because the proof uses an unproven theorem, it isn’t logically valid.”
“Th-that’s impossible….”
The solid common sense she had never once questioned until now.
At this very moment, it was silently crumbling away.
“Ah, so this is why university mathematics is completely different from high-school mathematics….”
Eun-chae unni trembled.
Looking at that expression, I felt strangely rewarded for teaching her.
“All right, then. I’ll show you how to prove it rigorously for real!”
Excited, I firmly adjusted my grip on the board marker.
✒️✒️✒️
Scritch-scritch-.
I wiped the whiteboard clean and firmly adjusted my grip on the board marker.
“First, we’re going to think of the number 0.9999… as a sequence, like this.”
Tick-tick-.
I wrote the equations on the board.
a₁ = 0.9
a₂ = 0.99
a₃ = 0.999
a₄ = 0.9999
…
aₙ = 1 − 10⁻ⁿ
“Look. We get a sequence that continues as 0.9, 0.99, 0.999, 0.9999…, and so on, right?”
“Oh, yes. One 9 gets added at a time.”
“Exactly! So 0.9999… can be regarded as the value this sequence reaches as it continues forever—in other words, its limit.”
I wrote the goal in large letters on the board.
[ 보이고 싶은 것! : lim aₙ = 1 ]
“Then let’s use the epsilon-delta definition we learned earlier to prove that this sequence really converges to 1.”
I turned to Eun-chae unni and asked her a question.
“What was epsilon again? The thing I explained in my own words!”
“Um… the tolerance?”
“Ding-ding-ding! Correct.”
I gave Eun-chae unni a thumbs-up.
Since she was being praised, the corners of her mouth lifted a little.
“Epsilon is the tolerance. So no matter how small a number you choose, all we have to show is that we can make the difference between the sequence aₙ and its limit value, 1, smaller than that error.”
“Smaller than the error….”
“Yep. Let me show you with an example. Suppose you throw an error of 0.001 at me.”
I wrote the number on the whiteboard.
“Then starting with the third term, a₃ = 0.999, the difference from 1 fits neatly within 0.001, right?”
“Oh, it really does. 0.999 and 1 differ by exactly 0.001… and after that, the difference keeps getting smaller….”
“Right! Now let’s say you throw an even tougher error at me: 0.000001.”
I wrote another number.
“Then starting with the sixth term, a₆ = 0.999999, every term after that falls within the error.”
“Ooh….”
“Now let’s say you throw an absurdly tiny number at me, with ten billion zeros after the decimal point.”
I pressed my thumb and index finger together to emphasize ‘absurdly tiny.’
“Even if you present an error that small, every term after the ten-billionth term of aₙ will slip neatly inside that error.”
“Hmm… I suppose it would?”
“That’s how it works. No matter what positive number you bring me, I can always meet that error. Without a single exception!”
I smiled brightly.
“This is the logical expression of the ‘gets infinitely close’ you learned in high school. You can think of it as translating the vague phrase ‘gets close’ into mathematical language.”
“Ah-ha…!”
A light of understanding flashed in Eun-chae unni’s eyes.
“All right, then let’s use this to prove it.”
Tick-tick-.
I neatly wrote the proof on the board.
………………………………………………………
Choose ε > 0 arbitrarily.
Choose a natural number N such that 10⁻ᴺ < ε.
(Specifically, a natural number N satisfying −log₁₀(ε) < N)
Then, when n ≥ N,
|aₙ − 1| = 10⁻ⁿ ≤ 10⁻ᴺ < ε
Therefore, by the definition of a limit,
lim aₙ = 1
∴ 0.9999… = 1
………………………………………………………
“Look, unni. This line here is the most important.”
I tapped the |aₙ − 1| = 10⁻ⁿ part.
“The difference between the sequence aₙ and 1 is exactly (0.1)ⁿ, right? As n gets larger, this gets smaller and smaller.”
“So no matter how small an ε you bring me, we can always find an N that makes it smaller than that.”
“So… no matter what error you throw at me, we can always find a point where it becomes smaller than that error….”
“Exactly! That’s it!”
“And since it satisfies the definition of a limit….”
Eun-chae unni’s voice grew clearer and clearer.
“The sequence aₙ converges to 1. Therefore, 0.9999… equals 1!”
“Correct! Perfect, unni!”
Clap-clap-clap-.
I clapped excitedly.
“Wow….”
Eun-chae unni stared blankly at the whiteboard.
“This… is a real proof. It’s completely different from my method earlier.”
“Right? The method you already knew assumed from the start that ‘0.9999… is a number on which arithmetic can be performed.’ But this method proves it using only the definition of a limit we discussed earlier, without making that assumption.”
“Ah-ha, so there’s no circular reasoning either… Wow, that’s seriously amazing.”
Eun-chae unni nodded in admiration.
Then she suddenly pulled me into a tight hug.
“Our adorable professor, you’re the best! I completely understand it now!”
“Mmph… Unni, I can’t breathe…!”
Rub-rub-rub-.
Eun-chae unni rubbed my cheeks again.