Is the procedure for proving $A = B$ the same as for proving $A \iff B$?

Question

To give a specific example, given a problem such as:

Prove that $\lim\limits_{x \rightarrow 0^+} f(1 / x) = \lim\limits_{x \rightarrow \infty} f(x)$,

The solution in my textbook has the form:

1. Suppose the $\delta, \epsilon$ definition of $\lim\limits_{x \rightarrow 0^+} f(1 / x)$ holds. Then the $N, \epsilon$ definition of $\lim\limits_{x \rightarrow \infty} f(x)$ also holds.

2. Conversely, suppose the $N, \epsilon$ definition of $\lim\limits_{x \rightarrow \infty} f(x)$ holds. Then the $\delta, \epsilon$ definition of $\lim\limits_{x \rightarrow 0^+} f(1 / x)$ also holds.

This format appears to be the same as the proof for an if and only if statement. Is it correct to say that proving equality statements and if and only if statements can be approached the same way? Also, can anyone recommend a reference for approaching different types of proofs? I am not looking for a comprehensive reference on logic, but instead a concise guide that would answer questions similar to this one.

Answer 1

Well, the book is really proving:

$\lim_{x\to 0^+} f(1/x)=L\iff \lim_{x\to\infty}f(x)=L$.

It's true in general that $A=B$ is equivalent to the statement:

For all values $L,$ $$A=L\iff B=L$$

This is more complicated than saying that two values are equal. What it is saying is that if either limit exists, then the other limit exists and they are equal.

That's the key. So there is an "iff or only if" part.

$A$ exists if and only if $B$ exists, and then they have the same value.

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Another time is when $U$ and $V$ are sets. Then $U=V$ is equivalent to:

For all $x,$ $x\in U$ if and only if $x\in V.$

So again, you prove both cases.