Convergent sequences are bounded

Proposition. Any convergent sequence in a metric space is bounded.

Proof sketch. If $x_n\to a$, then for $\varepsilon=1$ there exists $N$ such that $x_n\in B(a;1)$ for all $n\ge N$. The finitely many initial terms $\{x_1,\dots,x_{N-1}\}$ lie in some closed ball around $a$ of radius $r:=\max\{d(x_1,a),\dots,d(x_{N-1},a),1\}$. Hence all terms lie in $B'(a;r)$, proving boundedness (in the sense of bounded sequences ).