I want to prove the sequence a(n) = n diverges, directly, without the aid of any theorems. Naturally, I try to prove this by contradiction. Here's my attempt: Let L be a real number such that a(n) converges to L. Then for all e>0, there exists a natural number N s.t. any n>N implies d(a(n) - L) < e. So I say fine, fix that N and pick e=(1/2). Pick n such that n is the next natural number that is greater than N+L. Then d(a(n) - L) = d(n - L) > d(N+L-L) = N > (1/2). My question is did I do anything illogical by the way I picked my e and fixed the N? Also, since L is fixed, are my inequalities justified? I'm still getting used to the importance of the order of quantifiers and such.