Suppose $\lim x_{n}=x$ where $x\ne0$. Prove that the sequence $(-1)^{n}x_{n}$ is not convergent
My goal is to find two subsequences which converge to different values. Given the subsequence $y=(-1)^nx_n$ such that $n$ is odd gives the sequence $-x_n$. Now since the $\lim x_n=x$ then $\lim (-1)x_n=-x$ likewise given the subsequence $z=(-1)^nx_n$ such that $n$ is even gives the sequence $x_n$ thus $\lim x_n=x$. Therefore we have two subsequences which converge to different values so the sequence is divergent.
I feel I am missing some major key points any help would be appreciated