I'm having trouble with this question:
Let $V$ be an infinite dimensional vector space.
Prove that there exist subspaces $U_1,U_2,\dots$ of $V$ with $U_{n+1}\neq U_n$ for all $n\in\mathbb{N}$, such that $U_1\supset U_2\supset\cdots$ and $U_n$ is infinite dimensional for all $n\in\mathbb{N}$.
Thanks for your help.
Best Answer
Let $v_1,v_2,\dots$ be infinitely many linearly independent vectors in $V$.
Define $U_n:={\rm span}(v_n,v_{n+1},v_{n+2},\dots)$.