Good! Minced meat for NCA!
Here's a topological proof I found. It's much less direct but maybe the idea can be used elsewhere! Let's equip with the Euclidean metric. Note that is dense in . Let . Note that is not dense in .
If is invertible, then the theorem is trivial with .
Therefore suppose . Let . It's clear that is dense in since is dense in . If , then and therefore is dense in which is false. Therefore .