From what I can understand, he constructs a list of vectors in $U$ such that they are all linearly independent and since the length of linearly independent list cannot be greater than the length of the spanning list, the linearly independent list is finite thus proving U is finite dimensional.
My question: How does this idea of being able to construct a finite lineally independent list in $U$ show that $U$ is finite dimensional?
My thinking: For my other questions on this site, I usually include how I thought about it however for this I really don't know/have any ideas. I feel like I might be misinterpreting his proof somehow. I've seen the answer to this question Prove that "Every subspaces of a finite-dimensional vector space is finite-dimensional" however I still can't answer my question.