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Prove that Every Convergent Sequence is Bounded and Provide an Example of a Bounded Sequence that is Not Convergent

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Introduction

In mathematics, sequences are fundamental tools in analysis. Two key concepts related to sequences are convergence and boundedness. In this assignment, we will:

1. Prove that every convergent sequence is bounded.
2. Provide an example of a bounded sequence that is not convergent.

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1. Proof: Every Convergent Sequence is Bounded

Statement:

If {a_n} is a convergent sequence, then it is bounded.

Proof:

Let {a_n} be a sequence that converges to some real number L. By the definition of convergence, for every positive number epsilon (ε > 0), there exists a positive integer N such that:
|a_n – L| < ε for all n ≥ N.

Let us choose ε = 1.
Then, there exists an integer N such that for all n ≥ N:
L – 1 < a_n < L + 1.
This tells us that all terms from the N-th term onward lie within the interval (L – 1, L + 1), meaning they are bounded.

Now, define:
M = max(|a_1|, |a_2|, …, |a_{N-1}|, |L| + 1).

Since M is the maximum of a finite number of real numbers, it is finite.

We now claim that {a_n} is bounded by M. For any n:

• If n ≥ N, then |a_n| < |L| + 1 ≤ M.
• If n < N, then |a_n| ≤ M by construction.

Thus, for all n, |a_n| ≤ M. This shows that the sequence {a_n} is bounded.

Conclusion:

Every convergent sequence is bounded.

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2. Example of a Bounded Sequence that is Not Convergent

Consider the sequence {a_n} defined by:
a_n = (-1)^n.

The terms of this sequence are:
a_1 = -1, a_2 = 1, a_3 = -1, a_4 = 1, and so on.
The sequence alternates between -1 and 1.

Boundedness:

For all n, |a_n| ≤ 1. Thus, the sequence is bounded.

Non-Convergence:

For a sequence to converge, its terms must get arbitrarily close to a single real number L as n becomes large. However, in this sequence:

• The terms keep alternating between -1 and 1.
• There is no single value L to which the sequence converges.

Thus, the sequence {(-1)^n} is bounded but not convergent.

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Conclusion

In this assignment, we proved that every convergent sequence is bounded. We also provided the sequence {(-1)^n} as an example of a bounded sequence that does not converge. This demonstrates that while all convergent sequences are bounded, not all bounded sequences are convergent.

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