Cauchy Sequence is Bounded/Real Numbers/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

Every Cauchy sequence in $\R$ is bounded.


Proof

Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.

Then there exists $N \in \N$ such that:

$\size {a_m - a_n} < 1$

for all $m, n \ge N$.


In particular, by the Triangle Inequality, for all $m \ge N$:

\(\ds \size {a_m}\) \(=\) \(\ds \size {a_N + a_m - a_N}\)
\(\ds \) \(\le\) \(\ds \size {a_N} + \size {a_m - a_N}\)
\(\ds \) \(\le\) \(\ds \size {a_N} + 1\)

So $\sequence {a_n}$ is bounded, as claimed.

$\blacksquare$