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Introduction

A Cauchy sequence is a sequence where the terms get arbitrarily close to each other as the sequence progresses. One important property of Cauchy sequences is that they are always bounded. In this assignment, we will provide a formal proof to show that every Cauchy sequence of real numbers is bounded.

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Definition of a Cauchy Sequence

A sequence {a_n} is called a Cauchy sequence if for every epsilon > 0, there exists a positive integer N such that for all m, n ≥ N:

|a_m - a_n| < epsilon.

This means that as the sequence progresses, the difference between any two terms (after some point) becomes arbitrarily small.

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Theorem:

Every Cauchy sequence of real numbers is bounded.

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Proof:

We need to prove that if {a_n} is a Cauchy sequence, then the sequence is bounded.
A sequence {a_n} is bounded if there exists a positive real number M such that:

|a_n| ≤ M for all n in N,

where N is the set of natural numbers.

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Proof by Construction:

Since {a_n} is a Cauchy sequence, let us choose epsilon = 1.
By the definition of a Cauchy sequence, there exists a positive integer N such that for all m, n ≥ N:

|a_m - a_n| < 1.

Step 1: Show that the Tail of the Sequence is Bounded

For all n ≥ N, choose m = N. Then:

|a_n - a_N| < 1 for all n ≥ N.

This implies:

|a_n| < |a_N| + 1 for all n ≥ N.

Let M_1 = |a_N| + 1. Thus:

|a_n| ≤ M_1 for all n ≥ N.

Step 2: Bound the Initial Terms

Consider the first few terms of the sequence, from a_1 to a_{N-1}. These are a finite number of terms, so we can find the maximum absolute value among them. Let:

M_2 = max(|a_1|, |a_2|, ..., |a_{N-1}|).

Step 3: Combine the Bounds

Let:

M = max(M_1, M_2).

This ensures that:

|a_n| ≤ M for all n in N.

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Conclusion

We have shown that every Cauchy sequence of real numbers is bounded. This property ensures that such sequences behave in a controlled manner, even before we determine whether they converge. The boundedness of Cauchy sequences is an important result in mathematical analysis and is essential for understanding the completeness of the real numbers.