(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Theorem 1.4: Show that every Cauchy sequence is bounded.

2. Relevant equations

Theorem 1.2: If [itex]a_n[/itex] is a convergent sequence, then [itex]a_n[/itex] is bounded.

Theorem 1.3: [itex]a_n[/itex] is a Cauchy sequence [itex]\iff[/itex] [itex]a_n[/itex] is a convergent sequence.

3. The attempt at a solution

By Theorem 1.3, a Cauchy sequence, [itex]a_n[/itex], is a convergent sequence. By Theorem 1.2, a converging sequence must be bounded. Therefore, every Cauchy sequence is bounded.

I was just flipping through the textbook that my Analysis class will be using, "Introduction to Analysis" by Edward D. Gaughan, reading through Chapter 1. I noticed this theorem was left to an exercise, but I thought it was a bit too obvious of an answer as these two theorems in the Relevant equations were proven just before it. Is this really as simple as that?

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# Cauchy Sequences are Bounded

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