Proof

Let $X$ be a metric space.
Let $(x_n)_{n\in\mathbb N}$ be any convergent sequence in $X$.
It is to be shown that $(x_n)_{n\in\mathbb N}$ is a bounded sequence.
- From the proposition about convergent sequences together with limits in $X$ , it follows that the subset $\{x_n:~n\in\mathbb N\}\subset X$ is compact.
- Because compact subsets in metric spaces are bounded and closed, it follows that the subset $\{x_n:~n\in\mathbb N\}\subset X$ is bounded.
- By definition of bounded sequences, $(x_n)_{n\in\mathbb N}$ is a bounded sequence.
  ∎

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Forster Otto: "Analysis 2, Differentialrechnung im $\mathbb R^n$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984