Q:

Continuity of $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$

I have a question about a proof in my analysis textbook. 
They show that if $E$is a banach space, then $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$ is continuous by first showing that it is continuous in $I$ (the identity). Then they argue as follows: Let $U \in GL_c(E)$, let $r = \lVert U^{-1}\rVert^{-1}$, and let $V \in B_r(U)$. First, they write $V^{-1} = (U^{-1}\circ V)^{-1} \circ U^{-1}$. And say that now we can use the continuity of $J$ in $I$ to say that $J$ is continuous in $U$. 
I tried showing that $J$ is continuous in $U$ as follows.
$$\begin{align*}\lVert U^{-1} - V^{-1}\rVert  &= \lVert U^{-1} - (U^{-1}\circ V)^{-1} \circ U^{-1}\rVert\\ &= \lVert (I - (U^{-1} \circ V)^{-1}) \circ U^{-1}\rVert\\
            &\leq \lVert I - (U^{-1} \circ V)^{-1}\rVert\cdot\lVert U^{-1}\rVert\end{align*}$$
Well now I guess I have to use that I is continuous to get $||I - (U^{-1} \circ V)^{-1}||$ arbitrary small. But, I only have that $|| U - V || < r$. Could anyone point me into the right direction?

A:

Fix $\varepsilon>0$. By continuity in $I$, we can find $\delta$ such that if $\lVert I-A\rVert\leq \delta$ then $\lVert I-A^{-1}\rVert\leq\varepsilon$. So we have so see when $\lVert I-U^{-1}(U+H)\rVert\leq \delta$, i.e. $\lVert I-U^{-1}H\rVert$. It happens if $\lVert H\rVert\leq \frac{\delta}{\lVert U^{-1}\rVert}=\delta r$.

