To prove that there exists another subsequence of $(a_n)$ that converges, we need to carefully analyze the given conditions. Here's a structured approach:

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Step 1: Restating the Problem and Conditions

1. $(a_n)$ has two subsequences, one converging to $1$ and the other to $-1$.
2. For all $n$, $|a_{n+1} - a_n| < 1$.
3. We aim to prove that there exists another subsequence of $(a_n)$ that converges.

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Step 2: Key Observation

The two subsequences converging to $1$ and $-1$ imply that the sequence $(a_n)$ does not converge as a whole, but it oscillates indefinitely between these two values. However, due to the condition $|a_{n+1} - a_n| < 1$, the sequence cannot make arbitrarily large jumps. This bounded difference condition ensures some level of "local continuity" in the sequence's behavior.

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Step 3: Using the Tip: Infinite Points in $(-1 + \epsilon, 1 - \epsilon)$

• For any small $\epsilon > 0$, define the interval $I_\epsilon = (-1 + \epsilon, 1 - \epsilon)$.
• Since $(a_n)$ has subsequences converging to $1$ and $-1$, it must repeatedly enter and leave the interval $I_\epsilon$. Why? Because it cannot stay entirely outside $I_\epsilon$ (as that would contradict convergence of subsequences to $1$ and $-1$).
• Furthermore, due to the bounded difference condition $|a_{n+1} - a_n| < 1$, the sequence cannot skip over $I_\epsilon$ entirely. This implies there are infinitely many points of $(a_n)$ within $I_\epsilon$.

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Step 4: Extracting a Convergent Subsequences

1. Compactness Argument: The interval $[-1, 1]$ is compact. Any sequence contained within a compact set has at least one convergent subsequence (Bolzano-Weierstrass theorem).
2. The infinitely many points of $(a_n)$ within $I_\epsilon$ form a subsequence.
3. Since $I_\epsilon$ excludes the points $1$ and $-1$, any subsequential limit of this subsequence must lie strictly within $(-1, 1)$.
4. Therefore, this subsequence converges to some limit $c \in (-1, 1)$.

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Step 5: Conclusion

We have demonstrated that there exists at least one other subsequence of $(a_n)$ that converges to a limit $c \in (-1, 1)$.

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Would you like details on any part of the proof or further elaboration on the compactness argument? Let me know!

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Related Questions

1. What is the Bolzano-Weierstrass theorem and how is it applied here?
2. How does the bounded difference condition $|a_{n+1} - a_n| < 1$ prevent the sequence from skipping over regions?
3. Can you provide examples of sequences with similar properties?
4. What are the implications of having multiple convergent subsequences in terms of the original sequence's behavior?
5. How does the result change if $|a_{n+1} - a_n|$ is unbounded?

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Tip:

When dealing with sequences and their subsequences, always analyze their behavior in compact sets and use the Bolzano-Weierstrass theorem as a key tool for proving convergence.