Xueba, I beg you to go and escort him quickly!

Suddenly, Luo Yingnan was embarrassed!These math teachers are embarrassing!

They thought Xu Heng would not!

They want to help Xu Heng enthusiastically...!

They also said that Xu Heng can't do it either!

what……

So embarrassing!

Embarrassing to the extreme, isn't it...

"I really want to find a mouse hole to go in."

Xu Heng smiled, "It doesn't matter what anyone says, as long as they are right! Teacher, please explain it to me."

Luo Yingnan, "Ah..."

I thought it was showing off, but I didn't expect it to be a trick...

Hey……

It's hard to say!

But since Xu Heng has spoken, she has nothing to tweak.

"Our teachers studied for a while, and finally agreed to this proof process."

"To give a strict proof of the fact that 1=0.999 is cyclic, we first need to understand the method of constructing real numbers from rational numbers. This construction process will enable us to understand irrational numbers more deeply, instead of just staying on the intuitive level of infinite non-recurring decimals .”

As he spoke, Luo Yingnan began to write down the process on paper.

Suppose two non-empty sets of rational numbers A and B satisfy: A∪B is all rational numbers, and for any a∈A and b∈B, there is ab.

Then it is said that A and B constitute a Dedekind partition of the rational number set, referred to as partition, denoted as A/B.

Luo Yingnan explained, "From these two definitions, can you see the two meanings?"

"Level [-]: For any rational number a, it is either in A or in B, but not in both A and B..."

Yue Pingting murmured, "The second level is that every 18 rational numbers in A is less than any rational number in B?"

Luo Yingnan smiled, "Yes! That's right!"

"So, logically, the split A/B of the set of rational numbers may be one of the following four cases."

As she spoke, she wrote:

1: A has the largest number, B has no minimum number;

2: A has no maximum number, but B has a minimum number;

3: A has no maximum number, and B has no minimum number;

4: A has the largest number, and B has the smallest number.

Luo Yingnan, "But in fact, the fourth case cannot happen. Because if A has the largest number a and B has the smallest number b, we can know ab according to the definition of division. But (a+b)/4 is obviously also a rational number, and ..."

"..."

"Therefore (a+b)/2 is neither in A nor in B, which contradicts that A∪B is all rational numbers."

"In this way, the division A/B of the rational number set boils down to the following three situations..."

List them one by one.

What Luo Yingnan said was well-founded and methodical.

People have really studied it.

But Yue Pingting still subconsciously looked at Xu Heng.

She can have doubts about anything now, but she has no doubts about Xu Heng.

Her trust in Xu Heng can be said to be [-] percent, or even [-] percent.

But Xu Heng didn't say anything, and listened carefully.

Yue Pingting also continued to listen patiently.

"For the first case, we say that the split A/B determines the rational number a, for example, the example given above determines the rational number 1; for the second case, we say that the split A/B determines the rational number b, for example, the above given The example determines the rational number 0; and for the third case, that is, A has no maximum number, and B has no minimum number..."

"..."

"In this way, we get a strict definition of irrational numbers: Suppose A/B is a division of the rational number set, if there is no maximum number in A and no minimum number in B, then the division A/B is said to determine an irrational number c, c greater than Any rational number in A is also less than any rational number in B."

"E.g……"

She also gave an example!

For this proof, they can be said to have painstakingly.

Luo Yingnan used one sheet of straw paper after another, and it took more than half an hour to complete the entire proof process.

"..."

x=p/q≤1-1/q＜1-1/10(n次方)=0.999循环＜t.

Since xt, this means that x∈A.

"To sum up, we get A=C, so A/B and C/D are two identical divisions, so 0.999...9 (n nines)=t=9."

Luo Yingnan put down her pen, looked at these four or five pieces of paper, and she let out a long sigh, these are all her spoils just now!

She looked at Yue Pingting and Feng Nianshang, and then looked behind her. Before she knew it, many students came over.

Finally, Luo Yingnan looked at Xu Heng, "Student Xu Heng, what do you think?"

Several teachers also looked at Xu Heng in unison.

Another teacher touched Luo Yingnan twice, "Mr. Luo, you speak too fast, you should give student Xu Heng time to think."

They looked around and saw that many students had their faces covered.

"So... teachers, are you proving that 0.999 cycles equals 1?"

"How is this possible! Although I don't understand it, isn't this a big one?"

"I read it from the beginning, but I didn't understand it..."

All the students said it was difficult.

Yue Pingting and Feng Nianshang understood a little bit, but they still couldn't keep up with some thinking.

At this moment, Xu Heng frowned, "Your proof is wrong..."

Got a bug? ! ! !

Luo Yingnan, "???"

Other math teachers, "???"

Students, "???"

Everyone present was stunned!

All shocked!

Regardless of right or wrong, your thinking has really caught up?

Such a huge amount of proof, Xu Heng, can you keep up?

They don't believe it!

None of them believed it!

Including these math teachers!

"Student Xu Heng! Do you know? It took us more than half a month to prove it, you..."

One teacher said very excitedly.

But Xu Heng waved his hand and interrupted him, "But, there are still mistakes."

he,"!!!"

The veins on the forehead popped violently.

With that said, Xu Heng picked up the pen and found the mistake.

“错误在于显然地认为0.99…9（n个9）小于0.999循环小于等于1成立。”

"Your reason is to compare the size of each digit in turn from high to low, but this is not obvious. This property only exists after the 0.99...9 (n nines), 9 cycle has been defined. "

A hit!

Xu Heng directly refuted their mistake.

Luo Yingnan and the math teachers trembled instantly.

"This, this, this..."

The students looked at Xu Heng on the left, and Luo Yingnan and his math teachers on the right.

"My God! The gods are fighting!"

"What are they talking about? I can't understand it at all!"

"Hiss... what and what?"

"So forgiving! So brain-burning! My head can't take it anymore! It's horrible..."

"..."

Luo Yingnan froze in place, the corners of his eyes couldn't help twitching.

She seemed to realize the problem Xu Heng said.

But there are teachers who disagree.

"Student Xu Heng, you..."

He wanted to refute, but was stopped by Luo Yingnan.

"It's our authorities who have lost their minds!"

Luo Yingnan's face was embarrassing, although he didn't want to admit it, and he still admitted it in front of so many classmates, but——

"We were wrong! We were really wrong! This is an indisputable fact, Xu Heng, please enlighten me..."

After the teacher struggled fiercely in his heart, he also clenched his teeth and almost broke his back molars.

He was shaking with anger!

But he knows that in mathematics, a mistake is a mistake!

There is no room for excuses!

Otherwise, it's just a joke.

Swish swish!

All eyes were on Xu Heng.

Luo Yingnan, these teachers are!

The whole class is too!

Xu Heng picked up the pen and paper, and said slowly, "The cycle of 0.999 is equal to 1. Simply put, the decimal representation of real numbers is not unique. All real numbers, when they are finite decimals, have two decimal representations. When they cannot When represented as a finite decimal, it has only one decimal representation."

"The finite decimal here is a number that is all zeros starting from a certain digit after the decimal point."

Then Xu Heng wrote:

1 has only two decimal representations, 10 cycles and 1.000 cycles.

The 1.000 cycle represents the limit of the Cauchy sequence 1.0,1.00,1.000, 1, [-], ..., obviously his limit is [-].

The 0.999 cycle represents the limit of the CAUCHY sequence 0.9,0.99,0.999, 1, [-], ..., and the limit of this sequence is also [-].

"Let's deal strictly with real numbers."

"There are two ways to construct real numbers from rational numbers, Dedekind partitions, and Cauchy sequences."

"..."

"Let's look at the construction of real numbers first."