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Looking at this question, many students have no idea at all and don’t know where to start.

This is also very normal for them. After all, they have just started learning mathematical analysis. When encountering such a comprehensive problem, it is indeed difficult to find the solution to the problem at once.

There were also a few students who seemed to have some ideas and began to write on straw paper to try to answer the question.

After Ning Chen read the question, he first determined that it was a question he had never seen before, but he quickly found a way to solve the problem.

After all, Ning Chen is a contestant who has studied the entire Mathematical Analysis by himself, plus Advanced Algebra, Analytical Geometry, Elementary Number Theory, Abstract Algebra (Elementary), Mathematical Modeling and other With the blessing of several mathematical knowledge, Ning Chen had problem-solving abilities that were superior to those of mathematics majors.

First make a difference between f(x+t) and f(x), determine the range of this difference, and then according to the first form of Kronecker's theorem, let α1, α2,..., αn be any real numbers, θ1, θ2, ..., θn is a real number that is linearly independent on the integer ring...

Ning Chen quickly wrote his problem-solving process on the paper. He paused several times and thought briefly, but in the end he completed the entire proof process relatively smoothly.

So the sequence {tn} is what we want. In summary, the original proposition is proved.

After finishing the entire proof process in a relatively simple way, Ning Chen breathed a sigh of relief.

Speaking of which, Ning Chen was not satisfied with his proving speed.

It seems that I usually do too few questions, so I get stuck in the middle. If I were a mathematics major, I should have completed the entire proof process long ago.

But when Ning Chen looked up, he found that none of his classmates had raised their hands.

Everyone was either staring blankly at the questions on the big screen, or frowning and biting their pens, writing and revising on the straw paper, but no complete ideas emerged.

Seeing everyone's painful state, He Sheng on the podium looked relaxed.

Since everyone had answered the questions so smoothly before, in order to prevent the students from being too proud, He Sheng specially asked a question that was a little over the top to kill everyone's enthusiasm.

Seeing that no one raised their hands, He Sheng walked off the stage and began to patrol the aisles.

When he reached the last row, He Sheng saw that Ning Chen hadn't started writing and thought that Ning Chen didn't have any ideas.

But when He Sheng carefully looked at the process on Ning Chen's scratch paper, he was surprised to find that Ning Chen had already completed the proof of the problem.

Although Ning Chen's proof process was a bit simple, He Sheng knew that Ning Chen's idea was completely correct.

You...did you write this yourself?

Yes, teacher, but I don't know if I wrote it correctly.

In fact, Ning Chen was very confident in his own problem-solving ideas, but Ning Chen was not quite sure about the problem-solving format, so he tentatively asked He Sheng.

What you proved is correct, go and write it on the blackboard!

There was some uncontrollable excitement in He Sheng's words. After all, He Sheng could not have imagined that Ning Chen would be the first person in the classroom to answer this question.

There are more than a hundred students majoring in mathematics, but none of the auditors majoring in materials can do it quickly or well. This will inevitably make He Sheng feel incredible.

Ning Chen hesitated for a moment, then walked towards the podium.

Other students in the classroom also raised their heads and focused their attention on Ning Chen.

Seeing that the unknown student came to the podium, the mathematics students were all surprised and started talking among themselves.

Who is this student? Why did Teacher He let him come to the front?

I don't know, but since Teacher He asked him to go up, it seems that he really solved this problem.

It can't be a student who transferred from another major. I've never seen this person before.

If you transferred from other majors, you would not be able to learn so quickly. Maybe you are a student who has taken a leave of absence before and has already learned these courses.

Ning Chen didn't pay attention to other people's comments. He didn't even bring any scratch paper, so he picked up the chalk and started writing the process.

Because Ning Chen has completely memorized the entire proof idea in his mind, if he writes according to the scratch paper, the speed will be slower.

In addition, because the whole process has to be faced by all students, Ning Chen deliberately wrote the process in more detail, without omitting many intermediate processes as he just did on the draft paper.

Ning Chen's chalk writing is pretty good. Because he worked as a mathematics teacher in an education and training institution in his previous life, Ning Chen is very accustomed to using chalk.

Watching Ning Chen write the proof process of this question on the blackboard without any pause, the students in the classroom started talking one after another.

This person doesn't look simple. I feel that his process of writing proofs is very smooth. He can't be a new teacher in our major, right?

Although I can't understand the writing process, I still think it's very impressive.

Looking at this posture, I really don't know if four blackboards are enough for him.

There were also a few students who suddenly understood something after seeing the proof ideas written by Ning Chen.

So that's it. The entire problem-solving idea first uses the first form of Kronecker's theorem, and then discusses whether a1, a2, and a3 are linearly independent on the integer ring, and further proves that {tn} in the above conclusion is an upper bound. of……

I kind of understand it, but if I hadn't seen his way of thinking, I wouldn't have thought of this method at all. This requires too much basic mathematics. All situations must be listed, and there can't be any omission.

Give me two hours, I might be able to solve this question, but if I only have twenty minutes, I'm afraid I won't even be able to come up with the basic ideas.

When Ning Chen almost finished using two blackboards, he finally finished writing the process of this question.

It's very well written, go back.

He Sheng did not ask Ning Chen to explain his proof process to everyone, so Ning Chen did not stay on the podium and went straight to his seat.

Along the way, almost all the students in the classroom looked at Ning Chen, their eyes filled with various emotions.

Next, He Sheng explained to everyone the process of Ning Chen's writing.

After He Sheng's explanation, some students understood Ning Chen's problem-solving ideas, but this was only to an understandable level. It was still impossible for them to do it alone.

The classmate just now is a freshman majoring in materials science. Our students majoring in mathematics still need to continue to work hard.