Question

prove that - evey cauchy sequence is bounded.

Answer 1

SOLUTION

TO PROVE

Every cauchy sequence is bounded

EVALUATION

Let $\sf{ \: \{ u_n\}}$ be a Cauchy Sequence

We have to show that the sequence $\sf{ \: \{ u_n\}}$ is bounded

Let $\epsilon \: = 1$

Then there exists a natural number k such that

$\sf{ | \: u_m - u_n \: | < 1} \: \: \: for \: all \: m \: ,\: n \: \geqslant k$

$\therefore \: \: \sf{ | \: u_k - u_n \: | < 1} \: \: \: for \: all \: \: n \: \geqslant \: k$

$\sf{ \implies \: \: u_k - 1 < u_n < u_k + 1} \: \: \: for \: all \: \: n \: \geqslant \: k$

Let

$\sf{B = max \: \{ \: u_1, u_2, ....., u_{k - 1},u_{k} + 1 \: \} }$

$\sf{b = min \: \{ \: u_1, u_2, ....., u_{k - 1},u_{k} - 1 \: \} }$

$\therefore \sf{ \: \: b \leqslant u_n \leqslant B \: \: for \: all \: n \in \mathbb{N}}$

Hence the sequence $\sf{ \: \{ u_n\}}$ is bounded

Hence the proof follows

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