Where to publish a new proof of an old theorem? A few months ago I came up with a proof for an old theorem. After being excited for a moment, I then tried to find my proof in the literature. Since I did not find it, then I started to wonder if it was worth publishing it.
I asked a few people about journals that could publish something like this, and they gave me two recommendations:
(1) The Mathematical Gazette, http://www.m-a.org.uk/the-mathematical-gazette
(2)The Plus Magazine, https://plus.maths.org/content/about-plus
First I submitted to the Mathematical Gazette, and my article was rejected because according to the reviewer I was trying to prove something very simple using something much more complex (although I just used undergraduate level math).
Then I submitted to Plus, and it was also rejected by the editors (it probably doesn't fit well with their magazine). 
Do you have any suggestions? Thanks. 
 A: A new proof of an old theorem, especially using modern machinery, could well be within the scope of the Graduate Journal of Mathematics, which seeks to publish work either by graduate students or that would be of interest to graduate students. From their website:

The Graduate Journal of Mathematics is an electronic journal that publishes original work as well as expository work of general mathematical interest which add to the literature, have pedagogical value and help make more widely accessible significant mathematical ideas, constructions or theorems...One main aim of our journal is to help researchers in mathematics in general, and graduate students in particular, gain access to important ideas and communicate interesting mathematics.

Full disclosure: I am on the editorial board. And I'd be happy to receive a submission with a new proof of an old theorem.
A: This question, as stated cannot be answered. Everything depends on the theorem and on the proof, and this information you did not state.
For example, at least one Fields medal was awarded for a "new proof of an old theorem" (Selberg, 1950). A new proof can be published in principle in
any mainstream journal, if the theorem is important and the proof gives an important new insight.
A: If the old theorem is something commonly seen in an undergraduate math class (with the old demonstration), then this might be appropriate as a "Note" in the American Mathematical Monthly.  
What could happen if you submit it?  They may publish it.  The referee may give you a reference for it.  They may respond in the same way as the Gazette.  
What if the old theorem is not commonly seen in an undergraduate math course?  When you write a textbook on that area of math, you can include your new proof.  But if you think it unlikely you will write a textbook on this, then probably there is little prospect for publishing this.  Maybe if you make it known to the experts* then some day one of them may include it in their new textbook.
*Perhaps by posting somewhere on-line... 
A: Alternately, when you write a paper on a related topic (ie, which already develops the necessary machinery), you could perhaps include it somewhere in that paper?  I've seen this done numerous times.
