TITLE: Every matrix is a limit of a sequence of invertible matrices.
QUESTION [7 upvotes]: I'm trying to prove that every matrix $n\times n$ is a limit of a sequence of invertible matrices $n\times n$.
I know this intuitively, but I couldn't prove formally. I'm trying to prove this using the fact the set of the non-invertible matrices is closed. Am I on the right way?
Thanks in advance

REPLY [14 votes]: Let $A$ be your matrix, then $A-tI$ is invertible for all sufficiently small non-zero values of $t$, so just let $t$ run through any sequence of sufficiently small values approaching zero.