No. We can't conclude anything. Note that for each $m,n>1$, $mn$ is always a composite number. So we can define a sequence which takes a constant value $c_1$ on each composite number and another constant value $c_2$ on prime numbers. Then we can see that each of its subsequence of the form $(x_{mn})$ will converge to $c_1$ whereas the subsequence $(x_p)$ where $p$ runs over the prime number sets then $(x_p)$ will converge to $c_2$. This conclude that our sequence is not convergent.