Suppose $A \in B(H)$ where $H$ is a Hilbert space, and $B(H)$ denotes the collection of all bounded linear operators $A : H \rightarrow H$ on $H$, with the usual operator norm. Suppose furthermore that $A$ is invertible, i.e. there is $A^{-1} \in B(H)$ such that $A A^{-1} = id_H = A^{-1} A$.
I want to show that there is a neighbourhood of $A$ for which all operators in that neighbourhood are also invertible, i.e., there exists $\varepsilon > 0 $ such that for all $B \in B(H)$ with $ \| A - B \| < \varepsilon$, $B$ is invertible.
Not sure how to begin this argument. Have been told to consider using power series. A reference from a book would be appreciated as I am pretty certain this is a standard result. Thanks.