Convergent Sequence in Metric Space is Bounded

Theorem
All convergent sequences are bounded.

Proof
Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\mathbb{R}$.

Let $$x_n \to l$$ as $$n \to \infty$$.

We need to find $$K$$ such that $$\forall n \in \mathbb{N}: \left|{x_n}\right| \le K$$ from the definition of boundedness.

Since $$\left \langle {x_n} \right \rangle$$ converges, it is true that $$\forall \epsilon > 0: \exists N: n > N \Longrightarrow \left|{x_n - l}\right| < \epsilon$$.

In particular, this is true when $$\epsilon = 1$$

That is, $$\exists N_1: \forall n > N_1: \left|{x_n - l}\right| < 1$$.

By the Triangle Inequality, $$\forall n > N_1: \left|{x_n}\right| = \left|{l}\right| \le \left|{x_n - l}\right| < 1$$.

That is, $$\left|{x_n}\right| < \left|{l}\right| + 1$$.

So we set $$K = \max \left\{{\left|{x_1}\right|, \left|{x_2}\right|, \ldots, \left|{x_{N_1}}\right|, \left|{l}\right| + 1}\right\}$$

and the result follows.