I am trying to prove:

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> Let $g$ be a real valued function for all $x \in \mathbb{R}$. If $a \ne x$ is such that $\lim\limits\_{x\to a} |g(x)| = 0$, then $\lim\limits\_{x\to a} g(x) = 0$.
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Proof:

Let $x \in \mathbb{R}$, $g(x)$ be a real valued function, and $a \ne x$ be such that $\lim\limits\_{x\to a} |g(x)| = 0$. Let $\epsilon>0$. Then, $\exists \delta>0$ such that
\begin{equation\*}
 0<|x-a|<\delta \implies \left||g(x)|-0\right| = |g(x)| = |g(x)-0| < \epsilon
 \end{equation\*}
and we conclude that $\lim\limits\_{x\to a} g(x) = 0$.

Is this proof correct?