I'm suppose to use the definition of a limit of a sequence to prove the theorem that states:If limit(n approaches Infinity) of |A_n| = 0 then limit(n approaches Infinity) of A_n = 0.

If I showed that the "then" statement follows from the "if" statement by manipulating the "if" statement to arrive at the "then" statement, will I be proving the theorem? How would this prove the theorem to be true or am I not trying to prove the theorem to be true or false but rather that it is valid? (I'm sorry if I confused you with my questions ... I'm pretty sure I'm having trouble with the logic behind proofs and understanding what they're trying to ask me to do.)

Also, please give me a hint as to how to go about proving the theorem. Here is what I've done so far (and my accompanying notes for each step):

1. If lim|A_n| = 0 then lim A_n = 0 ; Theorem to be "proven".

2. lim|A_n| = 0 $\displaystyle \Rightarrow $ lim -A_n = 0 or lim A_n = 0 ; This is where I stopped since I don't know how to prove that the equivalent 2 limits equal zero and i'm not sure how proving them to be zero would prove that lim A_n = 0;