Let $(X,\Vert . \Vert)$ be a Banach space, assume that {${T_n}$} is a sequence of invertible operators in B(X) which converges to T $\in$ B(X). Suppose also that ${\Vert T_n ^{-1}\Vert}<1$, $\forall$ n$\in$ $\mathbb{N}$. $\\$ Show that T is invertible.
I don't even have an idea on how to go about this problem. Any help will be greatly appreciated. Thanks.
HINT Try to use the data you are given to conclude that for some $n$, we have $$\|TT^{-1}_n-1\|<1.$$ Because then by a standard argument, $TT_n^{-1}$ is invertible, and therefore $T$ is invertible with inverse $T^{-1}=T_n^{-1}(TT_n^{-1})^{-1}$.