Can someone please help me prove whether this sequence converges or not? I am having trouble figuring it out. Should I find some $\epsilon$ such that its greater than our sequence? Thank you for your time and help!

For the following sequence how do I show it converges by guessing the limit? Also, how can I verify the definition of convergence, or show it does not converge.

(1) Sequence $\{x\_k\}\_{k=1}^\infty$ in $\mathbb{R}$ given by $$x\_k = \left\{ \begin{array}{ll}
 10^{100} & \mbox{if $k =10^{1000}$};\\
 0 & \mbox{if $k\ne 10^{1000}$}.\end{array} \right.$$

$\text{Solution:}$ I believe this converges as $x\_k$ goes to 0 but I am not entirely sure.