Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series 
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*Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any $|\epsilon| < \epsilon_0$.

*Now suppose $S: V\to V$ is an invertible linear transformation and let $T$ be as in $(1)$. Show that there exists $\epsilon_0 > 0$ $($depending on $S$ and $T)$ so that $S + \epsilon T$ is invertible for any $|\epsilon| < \epsilon_0$.


For such a fundamental fact in the study of linear algebra I cannot find a proof of this anywhere... can anyone supply a proof or refer me to a place where I can find one?

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*A matrix is invertible iff the determinant is not equal to zero.

*Determinant is a continuous function of the entries of a matrix.

