Abstract thinking and proofs are why mathematics is so valued. But making proofs is not easy. It takes a lot of time and practice to just learn to read proofs, so it takes even longer to write them. Studying mathematics constantly trains you in reading and writing proofs. So do not try to run before you can walk. Be patient.
Read proofs (learning how to read, precedes learning how to write). Read the lecture notes. Pay special attention to examples or material that relate to the exercise that you are trying to solve.
Can you imitate the approach used in the lecture notes for your exercise?
Can you find tips on how to proceed or formalize parts of the proof you are trying to write?
Put the lecture notes on the side, are you able to reproduce the proof?
Start by writing the definitions of every object and property that appears in the problem. This is the way of introducing mathematical language. Sometimes it is also useful to write the negation of these definitions.
Proof types. Consider different ways of carrying out the proof: direct (A, indirect or proof by contradiction. Can you write the statement in indirect form? What would be a proof by contradiction?
The process. Writing a proof is a process
Write a first draft.
Read it critically: detect gaps and inconsistencies.
Think about how to fix these.
Revise and repeat until you produce the best version possible.
Sometimes, you may need to rewrite the proof entirely—this is normal!
Emotions. Writing proofs requires full focus. Emotions will generally unfocus you. So if an emotion arises like frustration or excitement, first calm down the emotion.
version date: 28th of March 2025 - AI was used to polish the text