Introduction

The Bolzano–Weierstrass Theorem is a fundamental result in real analysis. It ensures that every bounded sequence in the set of real numbers has at least one convergent subsequence. This theorem is crucial in the study of sequences, limits, and continuity.

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Statement of the Bolzano–Weierstrass Theorem

Theorem:
Every bounded sequence in the set of real numbers (R) has a convergent subsequence.

In other words, if a sequence {a_n} is bounded, then there exists a subsequence {a_{n_k}} that converges to some real number L, meaning:

lim (k → ∞) a_{n_k} = L

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Proof of the Bolzano–Weierstrass Theorem

Given:

A sequence {a_n} that is bounded.
This means there exists a positive real number M such that:

|a_n| ≤ M  for all n ∈ N

(Here, N is the set of natural numbers.)
Thus, all terms of the sequence lie within the interval [-M, M].

To Prove:

There exists a convergent subsequence {a_{n_k}}.

Proof:

We construct the convergent subsequence using the bisection method.

1. Divide the Interval:
   Since the sequence {a_n} is bounded by [-M, M], we divide this interval into two equal parts:

• Left half: [-M, 0]
• Right half: [0, M]

1. Choose the Part with Infinitely Many Terms:
   At least one of these halves must contain infinitely many terms from the sequence (since the sequence is infinite).

• Select this half and call it I_1.

1. Repeat the Process:
   Divide the selected interval I_1 into two equal parts.

• Again, select the part that contains infinitely many terms and call it I_2.

1. Continue Indefinitely:
   By repeating this process, we obtain a nested sequence of closed intervals:

   I_1 ⊇ I_2 ⊇ I_3 ⊇ ...

Each interval I_k has length M / 2^k after the k-th step.

5. Limit Point:
   According to the nested interval property, the intersection of all these intervals contains exactly one point. Let this point be L.
   This will be the limit of the convergent subsequence.
6. Construct the Subsequence:
   Now, choose terms from the sequence {a_n} such that:

• The first term of the subsequence lies in I_1.
• The second term lies in I_2, and so on.
• This construction ensures that the selected terms form a subsequence {a_{n_k}} that converges to L.

Thus, we have found a convergent subsequence from the original bounded sequence.

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Conclusion

The Bolzano–Weierstrass Theorem guarantees that every bounded sequence in the real numbers has at least one convergent subsequence. This result is important in analysis and is often used to prove properties related to compactness and continuity.