Cauchy Sequence is Bounded/Real Numbers/Proof 1
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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.
$\blacksquare$