Prove It

After observing that

               1  =  1
and         1 + 3  =  4
and         1 + 3 + 5  =  9
and         1 + 3 + 5 + 7  =  16
and         1 + 3 + 5 + 7 + 9  =  25

it seems easy to conclude that, for any positive integer n, the sum of the first n odd integers is n².