Convergent Real Sequence is Bounded/Proof 1

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:
 * $\exists M > 0: \forall n, m \in \N: \size {x_n - x_m} \le M$

Then for $n \in \N$, by the Triangle Inequality for Real Numbers:

Hence $\sequence {x_n}$ is bounded.