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).

Since all norms are equivalent here (you are in finite dimension), you are free to pick the one that is the most convenient. In particular, let $\lVert\cdot \rVert\colon \mathbb{R}^{k\times k}\to [0,\infty)$ be a the norm defined by $\lVert A\rVert = \max_{1\leq i,j\leq k} \lvert A_{i,j}\rvert$.
Since for all $n\geq 1$ we have $I_{k} = A_n\cdot A_n^{-1}$, we can apply the determinant to get
$$
1 = \det I_{k} = \det (A_n\cdot A_n^{-1}) = \det A_n \cdot \det A_n^{-1}
$$
for all $n\geq 1$, so that (as usual)
$$
\forall n\geq 1,\qquad \lvert \det A_n^{-1}\rvert  =  \frac{1}{\lvert \det A_n \rvert} \xrightarrow[n\to\infty]{} +\infty
$$
since $\det A_n \xrightarrow[n\to\infty]{} \det A = 0$, the determinant being continuous. But again, the determinant is a polynomial of fixed degree in the $k^2$ coefficients of the matrix.
Write $\det M$ as a polynomial $P$ in $k^2$ variables $(M_{i,j})_{i,j}$:
$$
\det M = P((M_{i,j})_{i,j}) = \sum_{S\subseteq [k]\times [k]} c_{S} \prod_{(i,j)\in S} M_{i,j} 
$$
Then $\lvert \det M\rvert \leq \sum_{S\subseteq [k]\times [k]} \lvert c_{S}\rvert \prod_{(i,j)\in S} \lvert M_{i,j} \rvert \leq 
\sum_{S\subseteq [k]\times [k]} \lvert c_{S}\rvert \lVert M\rVert^{\lvert S\rvert} \leq \lVert M\rVert^{k^2}\sum_{S\subseteq [k]\times [k]} \lvert c_{S}\rvert$,
i.e. there exists a constant $\gamma_k>0$ (only depending on $k$) such that
$$
\lvert \det M\rvert \leq \gamma_k\cdot \lVert M\rVert^{k^2}
$$
for every $M\in\mathbb{R}^{k\times k}$.
In particular,
$$
\lVert A_n^{-1}\rVert \geq \frac{1}{\gamma_k}\lvert \det A_n^{-1}\rvert^{1/k^2}\xrightarrow[n\to\infty]{} \infty
$$
