Let $\{p\_n\}$ be a sequence whose range is all of $\mathbb Q$. Prove that the set of subsequential limits of $\{p\_n\}$ in $\mathbb R$ is all of $\mathbb R$

My theorm that if $\{p\_{n\_i}\}$ as a subsequence of $p\_n$ converges, then it's limit is the subsequential limit of $p\_n$, but isn't this telling me that $\mathbb Q$ converges to $\mathbb R$? I'm thinking this because $\{p\_{n\_i}\}$ is a subsequence of $\mathbb Q$ and so it's also a subsequence of $\mathbb R$?