Metric space/Composition/Continuity in point/Exercise

Let ${}L,M,N$ be metric spaces, and let

$f:L\longrightarrow M\,\,{\text{ and }}\,\,g:M\longrightarrow N$

mappings. Suppose that ${}f$ is continuous in ${}x\in L$, and that ${}g$ is continuous in ${}f(x)\in M$. Show that the composition

$g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)),$

is continuous in ${}x$.

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