Prove that a set $S$ of vectors is linearly independent if and only if each finite subset of $S$ is linearly independent.
My progress: I have been able to prove that if $S$ is linearly independent then every subset of it is too.
How to show the converse?
I'm assuming $S$ to consist of subsets $S_1,\cdots S_n$ and since each are linearly independent the result follows.
Question: Is it ok to assume that $S_1,\cdots S_n$ are disjoint? If not, can someone tell me how to prove the result in that case?
Best Answer
OP's suggested approach is not helpful.
The key to this problem is the definition of linear independence of an infinite set: every linear combination of its elements with finitely many nonzero coefficients (not all of them zero) cannot yield the zero vector.