Pseudoinverse of Gaussian matrix converges to its transpose

Question

Let $X$ be an $n\times m$ matrix with iid $\mathcal{N}(0,1)$ entries. I would like to show that $$X^T\left(XX^T\right)^{-1}\rightarrow X^T$$ as $m\rightarrow \infty$, and $n$ is held fixed (and small). Showing that $XX^T\rightarrow I$ is easy, one can apply the strong law of large number per entry ($n^2$ entries, where $n$ is fixed). But since $X$ is not independent of $XX^T$, and the dimension of $X$ grows with $n$, I have difficulty showing the next step.