If $||B_N - B|| = 0$ as $N\to \infty$ for a bounded and invertible operator $B$, then does $||(B_N)^{-1} - B^{-1}|| = 0$ also hold as $N\to \infty$?

Question

Suppose $B:L(X,Y)$ is a bounded and invertible operator. Suppose we have another operator $B_N$ that approximates $B$ in the sense that as $N \to \infty$ it holds that $$ ||B_N - B|| = 0. $$ Then, does it also that hold that $$ ||(B_N)^{-1} - B^{-1}|| = 0, $$ as $N\to \infty?$