Convergent Real Sequence is Bounded/Proof 1

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

Let $l \in A$ such that $\displaystyle \lim_{n \mathop \to \infty} x_n = l$.

Then $\left \langle {x_n} \right \rangle$ is bounded.

That is, all convergent real sequences are bounded.

Proof
From Real Number Line is Metric Space, the set $\R$ under the usual metric is a metric space.

By Convergent Sequence in Metric Space is Bounded it follows that:
 * $\exists M >0: \forall n, m, \left \vert {x_n - x_m} \right \vert \le M$

Then for $n \in \N$, by the triangle inequality:
 * $\left \vert {x_n} \right \vert = \left \vert {x_n - x_1 + x_1} \right \vert \le \left \vert {x_n - x_1} \right \vert + \left \vert {x_1} \right \vert \le M + \left \vert {x_1} \right \vert$

Hence $\sequence {x_n}$ is bounded.