End of step symbol

Question

This is more of a style question.

We all know to end a proof with the good old QED (I use LaTeX's $\qed$ $\square$). I have a proof that is kind of long, and I was to put a delimiter to say "This is the end of a major part of the proof". It's not really something that I think should go in a Lemma or anything: it's just sort of the end a part.

EDIT: I use paragraphs right now, and that is really the most common way, I guess, but I'm kind of curious if anyone has other ways of marking pauses or "end of thought".

It's just a homework problem, and it really isn't long enough to even need to break up, but I was curious if anyone had any symbols/ways of doing this. (If your curious as to the problem, I'm proving a topology is metrizable by first proving that my function is a metric and then showing it is a metric for that topology.)

Answer 1

If a proof consists of proving properties (i), (ii), (iii) of an object say, I have encountered $\square_{(i)}$ and the like. However, I think it is much clearer and less confusing to the reader to simply group the text into paragraphs, each one ending in a clear statement what has been achieved and only mark the very end of proof suitably.

Proof. Yada yada ... This shows that my object has property (i).

In order to show that my object has property (ii), consider ... Hence property (ii) holds indeed.

Finally notice that property (iii) follows from ... completing the proof. $\square$

Similarly, if one wants to prove an equivalence, it is much mor common to group the proof into two paragragphs and introduce these with "$\Rightarrow$: " and "$\Leftarrow$: ", respectively, rather than add qed-like symbols at the ends of the parts.

By the way, in your case I am surprised that you are able to show that a function is a metric before you show that it is a metric for that space. Don't you define your function on $X\times X$ to begin with?