In Horn & Johnson 2012 "Matrix Analysis 2nd ed", page 85, they argue that the group of unitary matrices is closed, since for any convergent series unitary matrices, every matrix in the series being unitary implies that the limit is unitary. I feel this argument is slightly cyclic. Why does the argument hold?
Below, you find a screen shot of the relevant section, and I've highlighted the argument I don't understand.
screenshot from the book - relevant section
The matrix multiplication and adjugate are continuous map $\mathbb C^{n^2} \times \mathbb C^{n^2}\rightarrow \mathbb C^{n^2}$ and $\mathbb C^{n^2} \rightarrow \mathbb C^{n^2}$. Hence if $U =\lim_k U_k$ exists and the $U_k$ are unitary we have $$U^*U = (\lim_k U_k)^*(\lim_k U_k) = \lim_k (U_k^*U_k) = \lim_k I = I.$$