Geometry

# 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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