I am studying randomized SVD. However, many works give the error of randomized SVD like $$ \|A-QQ^TA\|_F\leq (1+\epsilon)\|A-A_k\|_F,\text{ and}\quad \|A-QQ^TA\|_2\leq (1+\epsilon)\|A-A_k\|_2 $$ where $A_k$ is the best rank $k$ approximation of $A$.
Intuitively, $\|A_k-(QQ^TA)_k\|_F$ and $\|A_k-(QQ^TA)_k\|_2$ should approach $0$ when $\epsilon\rightarrow 0$ where $(QQ^TA)_k$ is the best rank $k$ approximation of $QQ^TA$. Is this correct if the low rank approximation of $A$ is unique?