To prove that a Cauchy sequence is bounded, we need to show that if $\{a_n\}$ is a Cauchy sequence, then there exists some real number $M$ such that for all $n \in \mathbb{N}$, $|a_n| \leq M$.

Definition of a Cauchy Sequence:

A sequence $\{a_n\}$ is called a Cauchy sequence if for every $\epsilon > 0$, there exists an integer $N \in \mathbb{N}$ such that for all $m, n \geq N$, $|a_n - a_m| < \epsilon.$ In other words, as $n$ and $m$ get large, the terms of the sequence $\{a_n\}$ become arbitrarily close to each other.

Proof that a Cauchy Sequence is Bounded:

1. Start with the Cauchy condition: Since $\{a_n\}$ is Cauchy, for $\epsilon = 1$, there exists an integer $N \in \mathbb{N}$ such that for all $m, n \geq N$, $|a_n - a_m| < 1.$ This implies that for all $m, n \geq N$, the terms $a_n$ and $a_m$ are within 1 unit of each other.

2. Bound the terms from $n = N$ onward: Let $M_1 = \max\left( |a_1|, |a_2|, \dots, |a_{N-1}| \right)$. This is the maximum of the first $N-1$ terms, so the sequence is bounded up to $N-1$.

   For $n \geq N$, we can bound $|a_n|$ as follows:

   • For $n \geq N$, we know that for all $m \geq N$, $|a_n - a_m| < 1$.
   • In particular, consider $m = N$. Then for $n \geq N$, we have: $|a_n - a_N| < 1.$ Therefore, $|a_n| \leq |a_N| + |a_n - a_N| < |a_N| + 1.$ Thus, for $n \geq N$, the terms of the sequence are bounded by $|a_N| + 1$.

3. Combine the bounds: Now, the entire sequence $\{a_n\}$ is bounded. For $n \geq N$, we know that: $|a_n| \leq |a_N| + 1.$ For $n < N$, we already know that the terms $|a_1|, |a_2|, \dots, |a_{N-1}|$ are bounded by $M_1$.

   Therefore, the sequence is bounded by: $M = \max\left(M_1, |a_N| + 1\right),$ and for all $n \in \mathbb{N}$, we have: $|a_n| \leq M.$

Conclusion:

We have shown that the sequence $\{a_n\}$ is bounded by $M = \max\left(M_1, |a_N| + 1\right)$, where $M_1$ bounds the first $N-1$ terms, and $|a_N| + 1$ bounds the terms from $n \geq N$.

Thus, every Cauchy sequence is bounded.

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Would you like a more detailed explanation of any part of the proof? Here are some related questions:

1. What does it mean for a sequence to be convergent?
2. How does the concept of completeness relate to Cauchy sequences?
3. What is the difference between a bounded sequence and a convergent sequence?
4. Can a sequence be bounded but not Cauchy?
5. How do Cauchy sequences in other metric spaces behave?

Tip: The property that Cauchy sequences are bounded is key to understanding their behavior in a complete metric space. This is especially useful in real analysis and the study of limits.