[Math] Infinite dimensional vector space, and infinite dimensional subspaces.

linear algebravector-spaces

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 $U_n:={\rm span}(v_n,v_{n+1},v_{n+2},\dots)$.

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