If $L(\mathbb{R^n})^{−1}$ is the set of all invertible operators in $L(\mathbb{R^n})$, and if T ∈ $L(\mathbb{R^n})$ such that ||T || < 1
I've already proved that
(a) the series $\sum_{k=0}^{\infty} T^k $ converges,
(b) $I_n − T$ $\in L(\mathbb{R^n})$, and $\sum_{k=0}^{\infty}T^k = \frac{1}{I_n −T}$
Let the function $F$ : $L(\mathbb{R^n})^{-1}$ → $L(\mathbb{R^n})^{-1}$ be defined by $T$ → $T^{−1}$
Now I would like to show the following:
(i) The function F is continuous at $I_n$.
(ii) The function F is continuous on $L(\mathbb{R^n})^{-1}$
Form the series expansion you have obtained $||(I_n-T)^{-1}-I_n|| \leq \sum_{k=1}^{\infty } ||T^{k}|| =\frac {||T||} {1-||T||} $. Hence $||F(I_n-T)-F(I_n)|| \leq \frac {||T||} {1-||T||} $. Letting $||T|| \to 0$ we see that $F$ is continuous at $I_n$. If $T_k \to T \in L(\mathbb R^{n})^{-1}$ then $T^{-1}T_n \to I_n$ so $F(T^{-1}T_n) \to I_n$. Can you deduce from this that $F(T_n) \to F(T)$?