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Lately I have been getting into solving problems in some of the math journals I enjoy reading. More and more I find that solvers employ a theorem or identity that makes solving the problem much easier. Sometimes, that identity or theorem is one I am not familiar with.

For instance a few weeks ago I saw the Stolz-Cesaro Theorem used on a problem in the Fibonacci Quarterly. It was used in a very slick way, and was a theorem that up to that point I was unfamiliar with.

My question is: what are some of the best theorems, identities, inequalities, etc.. that the consummate problem solver should have at their disposal?

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closed as too broad by Lucian, Mike Earnest, user133281, Adam Hughes, Mark Bennet Dec 10 '14 at 22:42

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I am pretty sure there is a similar question on the site asking for such a list. Have you tried searching? $\endgroup$ – Mariano Suárez-Álvarez Dec 10 '14 at 20:32
  • $\begingroup$ Possible duplicate - math.stackexchange.com/questions/178940/… $\endgroup$ – Train Heartnet Dec 10 '14 at 20:38
  • $\begingroup$ Tychonoff's theorem, Cauchy's integral formula (and its related results), Zorn's lemma, Banach-Alaoglu theorem, Riesz representation theorem and contraction mapping principle are some big ones that have been really useful for me. $\endgroup$ – Cameron Williams Dec 10 '14 at 20:41
  • $\begingroup$ "(big-list) Please do not use this as the only tag for a question." $\endgroup$ – Thomas Dec 10 '14 at 20:55
  • $\begingroup$ I did a very quick search, and @Thomas: I wasn't sure what other tags I should use. I tried "identities" or "thoerems" but neither of those existed. But I will put those in. $\endgroup$ – FofX Dec 10 '14 at 20:57
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Here is newbies list,a staret pack of a kind:

-Mean value theorem

-Chain,addition,multiplication and other such rules for limits and derivatives

-Cauchy mean value theorem

-Rolles theorem

-Recursion theorem

-Variations of axiom of choice

-Darboux theorem

-Fundamental theorem of arithmetic

-Division algorithm

-Pigeonhole principle

-Binomial theorem

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  • $\begingroup$ I think Cauchy's, Rolle's and some other you didn't mentioned like Heine-Cantor could be packed in a "fundamental theorem on continuity" thing $\endgroup$ – servabat Dec 10 '14 at 20:49
  • $\begingroup$ It is because I did not know that good sir... :D $\endgroup$ – Vanio Begic Dec 10 '14 at 21:06

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