 
Summary: Tips for Writing Proofs
Drew Jaramillo
drewj@math.ucsb.edu
October 6, 2011
The following are some tips I think may be helpful for writing proofs. They come with the disclaimer
that they are my opinions (especially when it comes to style) and anything your professor says should
immediately override what is contained here. I do think they are good guides nonetheless since I
have been a grader and TA for a number of years as well as written my own proofs.
Tip 1. Write using complete english sentences
Many students especially when they start out, are in the mindset that math and english are totally
separate subjects. When it comes to proofs however, this is no longer really the case. A proof
is essentially a written argument: we start with axioms (or first principles) and using a series of
logical steps come to a conclusion. The logical steps may refer back to the axioms, a theorem, or
other general principles. It's fine to use symbols such as a A or {x, y} for sets or even to mean
"for all" and for "there exists" but one should keep the proof readable when using notation not
use notation as a substitute for saying what you want.
Tip 2. Make sure each step logically follows from the previous
All this means is that there should be no "gap" when reading through your proof. You should
always ask yourself when writing a proof "Why does this step follow from the last?" or "Am I using
just what is given or am I assuming something that is not in the problem?" This can be pretty
