Sequences

The converse of Theorem 1.2. (every convergent sequence is bounded) is not generally true. For example, observe the sequence $\left((-1)^n\right)_n$. The elements of these sequence consist only of $1$ and $-1$. It is bounded, yet not convergent. So we have a very great example of disproving a theorem by a counterexample. I love such examples because it is not important how big the theorem is, if you can find any counterexample then you can either throw that theorem into thrash bin or you can refine and redefine it. What we will do right now is refining and redefining. To do so, we need definition for monotonicity. ...

Intuition is a dangerous weapon Nearly every mathematician had very sharp and concise, yet sometimes dangerous intuition, which led mathematics into the very flawy statements. The theorems and even axioms in a system should be precise internally. I had talked about that topic in the page Gödel’s Proof. The definition (Definition 1.14) in section Analysis I Part 4 stated after the concept of convergence had been used without proof, relying only on intuition. Hopefully, it did not lead mathematics to flawy theorems, statements and etc. Why do we need such a solid definition of convergence? Well, it is because we need to prove them in general terms. Let us continue with boundedness and then move to the algebraic properties of limit? ...

Back then When I was in college, I used to solve a lot of sequence problems. I really liked the topic. It is something that I cannot depict right now, because I have only a few words in my mind. Whatever I write here cannot express the importance of the sequences in mathematics. Nevertheless, I will try to explain the topic while giving my understanding. While learning sequences it is important to build a bridge between the mathematics we know right know, and the mathematics that we will know in the future. ...