The following is basically Martin's argument but without messing with $\,\lim\sup\,$:
take any $\,\epsilon>0\Longrightarrow\,\exists\,N_\epsilon\in\Bbb N\,\,s.t.\,\,|x_n-x|<\epsilon\,\,,\,\forall\,n>N_\epsilon\,$ . Now, take $\,n>N_\epsilon\,$:
with $\,k\,$ a fixed positive constant. Well, now just make $\,n\to\infty\,$ ,and you'll get
and since $\,\epsilon\,$ was arbitrary we're done
Acclaration: As the above is a very common exercise at the start of limits of sequences, it may be it is given before $\lim\sup\,,\,\lim\inf\,$ and other beasts are studied, so perhaps the above approach is slightly more elementary.