Should I copy a short a proof from another paper or just refer to it?
A third option is to give your own proof.  Instead of copying it word-for-word, rewrite it in your own words and with notation that matches the rest of your paper, and include a citation of the original source.  

Don't do this by taking the source text and just changing a few words.  Instead, study it until you feel you understand the key idea, then set it aside and write your own proof.  Resist the temptation to just recite the original proof from memory.

This solves all your problems and more: 


it keeps the paper self-contained
it isn't plagiarism, since the source has been cited (to credit the idea) and the text hasn't been copied
it doesn't require quotation marks or other indication of verbatim quoting
it isn't copyright infringement, since the exact text wasn't used
it doesn't require explicit consent from the original author or publisher
you have gained a fuller understanding of how the proof works
your version of the proof will fit better in the context of your paper and your own writing style
you may be able to improve upon the original proof, or at least to make it better fit the way you think about the ideas involved
the reader now has two different proofs they can look at (yours and the original), which gives them some options in case they find something unclear in one of them.


At least in mathematics, it's most common to either do this, or else to  simply refer to the original.  It's very rare to actually copy a proof verbatim, and is usually only done when one wants to add critical or historical commentary on the proof itself.