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Subsequence Convergence


Which of the following sequences are subsequences of the sequence an=n2a_n=n^2? I. an=(2n)2II. an=2n2.\text{I. } a_n=(2n)^2\quad\text{II. } a_n=2n^2.

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

If the convergent sequence ana_n of positive numbers satisfies limnan2+an2=56,\lim_{n\to \infty} a_{n^2}+a_n^2= 56, what is the value of limnan\lim_{n\to\infty} a_n?

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

If a sequence ana_n satisfies limna2n=3\lim_{n\to\infty} a_{2n}=3, is it true that limna2n+1=3\lim_{n\to\infty} a_{2n+1}=3 as well?


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