You are still in the thinking inertia of constant or simple independent variable to dependent variable in high school. In fact, your attitude towards this matter is different.

Every time I use this language, what do I say at the beginning? How do I give a positive number ε? That's what it means.

In fact, this is a logical quantifier in mathematical logic. Using the symbol ε is a historical reason, and you can use other symbols. The most important thing is that it must represent any positive number.

Let me give you an example. You have to prove that-1 is less than all positive integers. You just have to prove that for any positive integer n, n+1>; 0 So N>- 1. According to the arbitrariness of n, we can know that-1 is smaller than any n. 。

The limit of sequence in Euclidean space is given in the form of ε in order to give a strict definition of limit. Don't think of ε as infinitesimal, it is just an arbitrarily given positive number, nothing else.

2.Xn represents the value of n term?

I don't understand your question. Xn generally refers to the item Xn corresponding to the index n.

3. The absolute value of xn -a is less than ε, that is to say, the value of n-a is less than ε.

Mathematics does not allow vague content to exist, so every item of it means itself. No subtext. You don't have to think too much. It means that the absolute distance between the value of the nth term and the constant a is less than ε. Actually, that's not the point. The more important premise is that you didn't type it. Where n is a positive integer greater than n, and n is written on the premise of any ε.

No matter what ε you give, I can find an n, which makes the absolute distance from all items to A after the first N+ 1 less than ε you give. The meaning of limit is that ε controls the distance from the infinite term to the fixed point A, and ε is any given positive number. Think about it, I can fit an infinite number of points with any positive number you give. Doesn't that mean that this series is infinitely gathered near this center?

4. See above.

It's normal for beginners not to understand for a while. It is normal for them to stay in a world without strict calculus for more than ten years, and it is normal for them to have this completely different thinking mode. Let me tell you a little story.

In the first grade of primary school, the teacher teaches the children who just graduated from kindergarten 1+ 1=2. The teacher said, "An apple+an apple is two apples ... so it is 1+ 1=2".

The children were overjoyed and said, "So 1 is an apple."

The teacher is anxious. "No, look, you turned the apple into a banana. A banana+a banana is two bananas. "

The children were puzzled: "1 Is it an apple or a banana?"

The teacher replied: "1 can mean apples and bananas, and anything that can be counted and added together will not produce other transformations."

The child was completely confused: "I understood at first, but now I have no idea what 1 is." Teacher, what is 1? "

In my freshman year, I studied * * *. The child who just graduated from junior high school asked the teacher, "Teacher, what is the element of * * *?"

Teacher: "Damn, asking so many questions. Just do the problem. Go away. Have a good college entrance examination. Don't be bored. "

And so on. Now you go to college, son.