The convergence of the sequence (A(n)) to a tells you that for all e > 0, there exists an N such that for all n > N, |A(n)-a| < e.

This last statement is equivalent to saying that a-e < A(n) < a+e, but you know that a > 0.

Now just take the e > 0 small enough such that a > e which implies a-e > 0 and thus A(n) > 0.