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# Topology

Explore geometric properties and spatial relations that are unaffected by continuous deformations, like stretching and bending. The next time you scan a bar code on a can of soda, thank a topologist.

# Subsequence Convergence

Which of the following sequences are subsequences of the sequence $$a_n=n^2$$? $\text{I. } a_n=(2n)^2\quad\text{II. } a_n=2n^2.$

If the sequence $$\{a_n\}$$ converges, what is the most we can say about how many subsequences of $$\{a_n\}$$ converge?

If the convergent sequence $$a_n$$ of positive numbers satisfies $\lim_{n\to \infty} a_{n^2}+a_n^2= 56,$ what is the value of $$\lim_{n\to\infty} a_n$$?

Consider a sequence $$a_n$$. If $$a_n$$ converges to $$x,$$ is it true that every subsequence of the $$a_n$$ has a further subsequence that converges to $$x$$?

If a sequence $$a_n$$ satisfies $$\lim_{n\to\infty} a_{2n}=3$$, is it true that $$\lim_{n\to\infty} a_{2n+1}=3$$ as well?

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