Show that the sequence of norms of inverses of a convergent sequence of matrices diverges to infinity. This is a question I found while working on the book "Analysis in Euclidean Spaces" by Ken Hoffman.  
Suppose $(A_n)$ is a sequence of invertible matrices from $\mathbb{R}^{k \times k}$ that converges to the matrix $A$. Show that if $A$ is not invertible, then $$\lim_{n \to \infty} \| A_n^{-1} \| = \infty.$$
I can easily show that if the sequence $(A_n^{-1})$ converges to some matrix $B$, then $B = A^{-1}$, but I don't know how to proceed if the sequence is not convergent (I can also prove the case where $A = \textbf{0}$, the zero matrix).

Using the $\max$ norm, we have 
$\tag1\vert \det A^{-1}_n\vert \leq k^2\Vert A^{-1}_n\Vert ^{k}. \ $ 
Also $\tag2\vert \det (A^{-1}_n)\vert =\frac{1}{\vert \det A_n\vert }\to \infty \ \text {as}\ n\to \infty $ 
because $A$ is not invertible and $\det$ is continuous.Thus, $(1)$ and $(2)$ imply that
$\tag3 \Vert A^{-1}_n\Vert \geq k^{-2/k}\vert \det A^{-1}_n\vert ^{1/k}\to \infty \  \text {as} \ n\to\infty$
