Let $\left \{ a_{n} \right \}$ be a sequence of real numbers. Then $\lim_{n \to \infty}a_{n}$ exists if and only if
(A)$\lim_{n \to \infty}a_{2n}$ and $\lim_{n \to \infty}a_{2n+2}$ exists
(B)$\lim_{n \to \infty}a_{2n}$ and $\lim_{n \to \infty}a_{2n+1}$ exists
(C)$\lim_{n \to \infty}a_{2n}$, $\lim_{n \to \infty}a_{2n+1}$ and $\lim_{n \to \infty}a_{3n}$ exist
(D)none of the above
They all seem correct to me. How to approach this ?
The first thing you should think about is the logic. It is a basic theorem about sequences that if the limit of a sequence exists, then the limit of any subsequence exists (and is the same). So if $\lim_{n\to\infty}a_n$ exists, then all the other limits mentioned exist and (A), (B), (C) are all true. I suspect that this is what you are thinking.
However, the question didn't say "if", it said "if and only if". So what you have to decide is, if (A) is true, does it follow that $\lim_{n\to\infty}a_n$ exists? And the same for (B) and (C).
Here is a hint. Can you devise a sequence $\{a_n\}$ for which $\lim_{n \to \infty}a_{2n}$ and $\lim_{n \to \infty}a_{2n+1}$ both exist but are not equal? If you can then (B) is ruled out; if you can prove that this is impossible, then (B) is the answer.
And a hint for (C). Start off thinking in the same way and consider the sequence $\{a_{6n}\}$.
