Convergent and Cauchy Sequences - Real Analysis | Lecture 2

In this lecture we introduce the notion of a convergent, Cauchy, and bounded sequence. We illustrate these definitions with a number of simple and illustrative examples. We also prove a chain of results that show that convergent sequences are Cauchy sequences and Cauchy sequences are bounded. From these results one can infer by transitivity that convergent sequences are also bounded. We demonstrate that this chain of results turns out to be quite useful in demonstrating that certain sequences are not convergent. Furthermore, Cauchy sequences will play a critical role in the following lecture to construct the real numbers from the rationals.

This course is taught by Jason Bramburger.

[marksman_]

Hello. I'm trying to learn real analysis from your video series, and had a question.
I know from my calculus class that x_n = ln(n) is not a convergent sequence, but from the video, it does seem to be a cauchy sequence (I couldn't prove it using the epsilon definition, but I observed that the derivative of the function ln(x+1) - ln(x) is negative, implying that the difference between successive terms always becomes smaller)
My question is, is every cauchy sequence convergent? Also, is every Bounded sequence cauchy and convergent?
Thank you

[peki_ooooooo]

Great! Can you recommend a analysis book for beginner?