# Prove that any finite subset of a linearly independent set is linearly independent

My problem: Prove that a set of vectors $S$ is linearly independent if and only if any finite subset of $S$ is linearly independent.

I tried like this:

Suppose S is LI.Then the vector $0$ cannot be expressed as a linear sum of all elements of $S$.

How it follows that a finite subset is also LI from this fact. I think $S$ can be finite or infinite.

This is a question from the book Linear Algebra - Friedberg et al.

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Well, you haven't even attempted one half of the problem. – Chris Eagle Jan 16 '13 at 15:04
@ChrisEagle I have started with the only if part. – Vinod Jan 16 '13 at 15:06
Is it necessary to use all elements of $S$ when creating the linear sum that is not supposed to be $0$? If $S=\{x,y\}$, is a linear sum $ax+by$ or must is be $x+y$ only? If the former, can I take $a=b=0$? How about $a=0, b\neq 0$ or vice versa? – Dilip Sarwate Jan 16 '13 at 15:08
@DilipSarwate linear sum $ax+by$ is possible. – Vinod Jan 16 '13 at 15:10

A set $S$ being linearly independent by definition means no non-trivial linear combination of elements in $S$ is zero. And a linear combination is a FINITE sum.Therefore if all finite subsets are LI, $S$ is LI.