Let (x_n) be a convergent sequence with limit a. Assume a>0. Prove that there is a threshold k such that x_n>0 for all n k.

What I have:

Let >0. Then there is a k such that for all n k, |x_n-a|< . Assume a>0. Then since k such that ( n k, |x_n-a|< . We know that 0<|[x_n-a|< which is a<|xn|< +a. Since a>0 we have |xn|>a>0.