Direct proof
For example, this statement can be solved with a direct proof: "if x and y are even integers, then x+y is an even integer." Since x and y are even, then we can say that x=2m and y=2n, where m and n are integers. (This is a lemma.) We can also say that m+n is an integer, because adding two integers gives an integer. (This is another lemma.) Then we can say that x+y=2m+2n=2(m+n). Since m+n is an integer, we can say 2(m+n)=2k, where k=m+n. We know that any integer times 2 is an even integer. We can then say that any two even integers added together give an even integer.