Permutations are hell for me

How do you prove that whatever permutation from S5 you take, there is n so that σ^n=e?

Let G be a finite group and g an element of G.  Consider the infinite sequence g^1, g^2, \ldots.  Since G is finite, there must be two elements of this sequence that are equal, say g^i = g^j with j>i.  Then by cancellation, g^{j-i} = e.

S_5 is a finite group.

Done.