(related to Proposition: Convergent Sequences are Bounded)

- 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.∎

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