Cauchy Sequence is Bounded/Real Numbers
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Theorem
Every Cauchy sequence in $\R$ is bounded.
Proof 1
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 $m, n \ge N$.
So for all $m \ge N$, we have:
\(\ds \size {a_m}\) | \(=\) | \(\ds \size {a_N + a_m - a_N}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {a_N} + \size {a_m - a_N}\) | Triangle Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds \size {a_N} + 1\) |
So $\sequence {a_n}$ is bounded, as claimed.
$\blacksquare$
Proof 2
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 $m, n \ge N$.
Note that for $m \le N$:
\(\ds \size {a_m}\) | \(\le\) | \(\ds \max \set {\size {a_1}, \size {a_2}, \dotsc, \size {a_N} }\) | |||||||||||||
\(\ds \) | \(<\) | \(\ds \max \set {\size {a_1}, \size {a_2}, \dotsc, \size {a_N} } + 1\) | |||||||||||||
and for $m > N$: | |||||||||||||||
\(\ds \size {a_m}\) | \(=\) | \(\ds \size {a_N + a_m - a_N}\) | |||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {a_N} + \size {a_m - a_N}\) | |||||||||||||
\(\ds \) | \(<\) | \(\ds \size {a_N} + 1\) | |||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\size {a_1}, \size {a_2}, \dotsc, \size {a_N} } + 1\) |
Hence for all $m \in \N$:
- $\size {a_m} < \max \set {\size {a_1}, \size {a_2}, \dotsc, \size {a_N} } + 1$
Therefore $\sequence {a_n}$ is bounded, as claimed.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Cauchy sequences: $\S 5.18$