Proof
A proof is a rigorous argument meant to convince readers that a theorem is true. They usually take the form of a step-by-step guide on how to reach a conclusion through the use of given assumptions. These assumptions can be axioms or other already-proven theorems.
There are many types of proofs: direct proofs, proofs by contradiction, proofs by exhaustion, proofs by induction, and the list goes on. Soon enough, you'll be well-acquainted with all of them!
Format
The format of a proof should be simple and clear. First, write "Theorem:" and state the theorem you intend to prove. Then, write "Proof:" and begin taking steps to reach the conclusion. After showing the theorem is true, end your proof with the end-of-proof symbol "\(■\)".
Proof: Let \(n\) and \(m\) be odd integers, where \(n=2k+1\) for some integer \(k\) and \(m=2j+1\) for some integer \(j.\)
Their sum must be:$$n+m$$ Substituting in the definitions of \(n\) and \(m\):$$(2k+1)+(2j+1)$$$$2k+2j+2$$$$2(k+j+1)$$ Since \(k\) and \(j\) are integers, \((k+j+1)\) must also be an integer, which we can rewrite as \(x\):$$2x$$ Now, \(n+m\) has been represented in the form \(2x\), where \(x\) is an integer. This is the definition of an even integer.
Thus, the sum of any two odd integers is even. \(■\)