# real analysis converging subsequences

I am having trouble finding a diverging sequence in R whose subsequences of the even indexed terms.

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Let $x_k$ be $k$ if $k$ is prime and $1$ otherwise. Then $\lim_{k \to \infty}x_{nk} = 1$ for all integers $n>1$. However, if $p_k$ is an enumeration of the primes, then we have $\lim_{k \to \infty} x_{p_k} = \infty$.
Let the sequence be such that $x_n=0$ if $n$ is a multiple of $2$ or $3$ and $x_n=1$ otherwise.