What proof techniques are specific to real analysis?

[SrVishi]

Every math major eventually learns logic and standard proof techniques. For example, to show that a rigorous statement $P$ implies statement $Q$, we suppose the statement $P$ is true and use that to show $Q$ is true. This, along with the other general proof techniques are very broad. A math major would soon come to realize that there are some nuances of proofs that vary among the different subjects. For example, in real analysis, a possible way to show that two real-valued objects are equal is to show that neither can be less than or greater than the other. What proof tips (could be as specific as you'd like) could you provide?

 

[tommyxu3]

There may be lots of different ways, and in special cases, may be more. Generally we all know we can prove $~Q\Rightarrow ~P$ to show $P\Rightarrow Q.$
In analysis, there are many tricky(?) estimations...which I always can appreciate...

 

[Mark44]

There may be lots of different ways, and in special cases, may be more. Generally we all know we can prove $~Q\Rightarrow ~P$ to show $P\Rightarrow Q.$

At first I thought you had written something that wasn't true in general, but after seeing the LaTeX you wrote, I understand what you meant.
Here's the corrected version:
prove $\neg Q\Rightarrow \neg P$ to show $P\Rightarrow Q.$

I used \neg for the logical negation symbol. You can also use \sim, which renders as a ~ character.

In analysis, there are many tricky(?) estimations...which I always can appreciate...

 

[tommyxu3]

Yes, I didn't mind that until your remind haha. Thanks a lot!