I have a question concerning subspaces of infinite dimensional vector spaces. Specifically given any infinite dimensional vector space V, how might one construct an infinite decreasing chain of subspaces? That is: V=V0[itex]\supseteq[/itex]V1[itex]\supseteq[/itex]... , where each Vi is properly contained in Vi-1. I know such chains must exist and I suspect that they should be easily constructed, however I am not familiar enough with infinite dimensional vector spaces to be confident with my attempts thus far. Thanks for any help given!