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.

Hence Convergent Sequence in Metric Space is Bounded can be applied directly.