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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.

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  • $\begingroup$ Yes, proving $A = B$ is the same as proving $A$ implies $B$ and $B$ implies $A$. $\endgroup$ – Wolfy Mar 19 '18 at 19:47
  • $\begingroup$ I think this is a good observation, but this question doesn't quite seem well-defined to me - are you paraphrasing? Statements 1 and 2 are doing a little more than just calculating a limit - they're also proving that these limits exist! $\endgroup$ – Billy Mar 19 '18 at 19:48
  • $\begingroup$ Well, no $A\implies B$ implies $A$ is a statement, but $A=B$ implies $A$ is a value. @Wolfy What do you think $A$ and $B$ are in this question? $\endgroup$ – Thomas Andrews Mar 19 '18 at 19:48
  • $\begingroup$ In this case the equality is equivalent to an if and only if statement because each side contains the premise that the limit exists. In general, they are not the same. Proving $x+y = y+x$ doesn't involve if and only if logic. There are books about approaching proofs; I'll let others recommend some. $\endgroup$ – Ethan Bolker Mar 19 '18 at 19:50
  • $\begingroup$ @ThomasAndrews I am trying to say that if we are to prove $A = B$ then we need to prove it from left to right and right to left. Is that not right? $\endgroup$ – Wolfy Mar 19 '18 at 19:50
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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.


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.

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  • $\begingroup$ Your comment about statements vs. values was helpful. The limit itself is a value, and the limit equaling $L$ is a statement. In addition to your second example about set equality, I have seen $U = V$ being proved by showing $U \subset V$ and $V \subset U$. Are you familiar with any sources that discuss such "proof techniques?" $\endgroup$ – tmakino Mar 19 '18 at 20:14
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    $\begingroup$ @tmakino A word about notation: $X\subset Y$ means "$X$ is a subset of $Y$ but not equal to $Y$"; use $X\subseteq Y$ to mean "$X$ is a subset of or is equal to $Y$". $\endgroup$ – Shaun Mar 19 '18 at 22:03

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