What is the reason to explore and study the Real Analysis Area?

Question

I repeated 4 analysis classes, this is the 5th time. I believe I'll pass, but I still can't understand Real Analysis. I can understand statistics, numerical analysis, Calculus I-II and even Topology, but can't get real analysis. I think to learn anything one must solve problems about that subject all by h{is,er}self, I didn't do that much in Real Analysis but still, I had to have some intuition after all this time. I can understand the definitions but it's not enough. It's like teacher explains it like you are playing mario but gives a homework similar to hardest game on world.

Level: Undergrad.

For Example: We start with Limit definiton o a sequence, then Corollary and Theorems come about bounded functions, Bolzano-Weierstrass... Etc, Then we study series and I start to lose why do we do this. So I alter my question to "What's the end goal of Analysis? To find functions' properties or how do they act in infinity?" If I knew that I would understand it better.

I quickly lose the track of our purpose. Are we on sequences? OR sequences of functions? Are we on series? I don't know why are we doing this. It's like someone said lets add this elements of a sequence together and call it series.

Any tips and tactics to comprehend this abstract beast of an area? Shortly, I can't "talk" Real Analysis, it's a very hard language for me I haven't cracked. Do you have any textbook recommendation?

Answer 1

Your posting gives so little information that the best possible answers may miss what you need to know, but I will hazard these two small points:

You need to understand:

• Proofs: How to judge whether are argument you see on the page is value and how to know what it says; and how to write valid proofs.
• The gaplessness of the real line. Suppose $A\cup B =\mathbb R$ and $A\ne\varnothing\ne B$ and $A\cap B= \varnothing$ and (important) every member of $A$ is less than every member of $B.$ Then there is a boundary point that is a member of either $A$ or $B$. A “gap” would be a pair of sets satisfying these four requirements but in which $(1)$ for every member of $A$ there is a larger member of $A$ and $(2)$ for every member of $B$ there is a smaller member of $B.$ The real line has no gaps.

  This seems to me a more intuitive way to explain the matter than talking about the "least upper bound property."