check convergence of sequence of real numbers

Question

Let $(x_n)_{n\in\mathbb N}$ be a sequence of real numbers such that the subsequences $(x_{2n})_{n\in\mathbb N}$ and $(x_{3n})_{n\in\mathbb N}$ converge to limits $K$ and $L$ respectively. Then does $(x_n)_{n\in\mathbb N}$ converge?

Answer 1

Not necessarily. Suppose that$$x_n=\begin{cases}1&\text{ if }2\mid n\text{ or }3\mid n\\0&\text{ otherwise.}\end{cases}$$Then the subsequences $(x_{2n})_{n\in\mathbb N}$ and $(x_{3n})_{n\in\mathbb N}$ converge to $1$, but the sequence $(x_n)_{n\in\mathbb N}$ doesn't converge.

Answer 2

1) If $K \not = L$, $(x_n)$ does not converge (why?).

A counter example:

2) Consider: $K=L$.

Let $p$ be an odd prime:

$x_n = 0$, for $n \not = p$.

$x_n=n$, for $n=p$.