Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent.
The Attempt at a Solution
I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors is linearly dependent if and only if there exists a finite subset of S that is linearly dependent.
First assume that a set S is linearly dependent. Then there exists a linear dependence relation among its vectors ##a_1s_1 + \cdots + a_ns_n = 0##, such that not all ##a_1,..., a_n## are zero. Since S is a subset of itself, there must exist a finite subset of S that is linearly dependent.
Second, assume that S has a finite subset S' that is linearly dependent. Since S' is a subset of S, this means that there exists a linear dependence relation among the vectors in S. Thus, S is also linearly dependent.
Does this prove the original statement? It seemed a little too easy.