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$\displaystyle \geqslant $k.

What I have:

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