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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