I think I read the question backwards originally (and I think some of the comments may also have done this). So I will give two answers, one for each reading.
About providing citations elsewhere to results proved here
The issue of when to cite things in mathematics is complex and delicate - and not well documented. Some advice can be found in the American Mathematical Society Ethical Guidelines:
The correct attribution of mathematical results is essential, both because it encourages creativity, by benefiting the creator whose career may depend on the recognition of the work and because it informs the community of when, where, and sometimes how original ideas entered into the chain of mathematical thought. To that end, mathematicians have certain responsibilities, which include the following:
To endeavor to be knowledgeable in their field, especially about work related to their research;
To give appropriate credit, even to unpublished materials and announced results (because the knowledge that something is true or false is valuable, however it is obtained);
Applying this advice in practice takes experience. Opinions about how to handle answers on math.SE vary.
Personally, I would treat an answer here the same as a personal communication: an email conversation or a conversation where someone explained something. In such circumstances, if I think the person should be credited, I might ask them if they would like to be thanked. If they would, I will mention their name in my paper.
For example, the acknowledgements section of a paper might include a line such as "I would like the thank John Smith for an explanation of the proof of Theorem 3", if the theorem turns out to be essentially an exercise, or a remark like "The following theorem is due to John Smith" just before the theorem, if the theorem turns out to be particularly interesting.
Others view math.stackexchange.com more as a form of publication, and less as a form of personal communication; they might also mention math.stackexchange.com in their acknowledgement. It's not a straightforward question, particularly, because it depends to some extent on how interesting the result is that you are using. In the question at Three definition of total boundness (for uniform spaces), the result is not likely to be particularly interesting as a research problem for experts in the field.
In the end, I would recommend discussing this with your PhD advisor or mathematical colleagues, to see what their opinion is.
About providing citations here to results proved elsewhere
For the question linked above, Three definition of total boundness (for uniform spaces), there is no reason to expect the person answering the question to include a reference for their answer. The question asked, "how to prove equivalence with the third" and that is what the answerer explained.
If you already know the citation when answering a question, there's no reason not to include it. If you look up an answer in a book to post it here, so that you are essentially just translating the book to this site, then a citation is certainly appropriate. But if you can give a good answer off the top of your head, without looking anywhere, there is no reason that you must or even should spend extra time looking up a book where the answer can also be found.
A key point is that math.stackexchange.com is not Wikipedia. On this site, we value personal knowledge (expert or amateur), and it is perfectly acceptable to answer a simple question by just explaining the facts of the matter. We do not expect to see answers peppered with "" marks. For questions that are about mathematics (not reference requests), people may prefer to see answers that focus on the math, rather than focusing on a list of references without directly answering the original question.