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Suppose $Q_kR_k \rightarrow QR$ as $k \rightarrow \infty$, where $Q_k, Q$ are orthogonal matrices and $R_k, R$ are upper triangular with positive diagonal entries, then would the uniqueness of the $QR$ decomposition imply that $Q_k \rightarrow Q$ and $R_k \rightarrow R?$ I need this detail for a proof, but I wasn't able to prove it.

It would suffice to show that if $Q_kR_k \rightarrow I$, then $Q_k \rightarrow I$ and $R_k \rightarrow I$.

Edit: If the limit of $Q_k$ and $R_k$ exist, then they must be $I$, but how would one show that these limits exist, if they do?

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Since the $Q_k$ are orthogonal, they must be bounded, so we can apply Bolzano-Weierstrass. As you mentioned in your edit, any convergent subsequence would have to converge to $I$.

If $Q_k$ does not converge to $I$, then we can take the subsequence of all $Q_k$ that are at least $\epsilon$ away from $I$. Bolzano-Weierstrass then tells us there would be a subsequence that converges to something other than $I$, which leads to a contradiction.

Convergence of $Q_k$ and $Q_kR_k$, along with invertibility of the $Q_k$, then implies convergence of the $R_k$.

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