Convergent Real Sequence is Bounded/Proof 1

Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.

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

Then $\sequence {x_n}$ is 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:

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

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

\(\ds \size {x_n}\)

\(=\)

\(\ds \size {x_n - x_1 + x_1}\)

\(\ds \)

\(\le\)

\(\ds \size {x_n - x_1} + \size {x_1}\)

\(\ds \)

\(\le\)

\(\ds M + \size {x_1}\)

Hence $\sequence {x_n}$ is bounded.

$\blacksquare$