I need help with the proof of this statement:
If $S=\big\{ \vec{v_1},...,\vec{v_{n}} \big\}$ is a set of linearly independent vectors then any non-empty subset of $S$ contains only linearly independent vectors.
I tried using the definition
$\vec{v_1},...,\vec{v_n}$ are l.i. $\iff \lambda_1\vec{v_1}+...+\lambda_n\vec{v_n}=\vec{o} \implies \lambda_1=...=\lambda_n=0$
But is this enough to conclude that, for example, considering a subset $S'=\big\{ \vec{v_1},...,\vec{v_{n-1}} \big\}$ for sure $\lambda_1\vec{v_1}+...+\lambda_n\vec{v_{n-1}}=\vec{o} \implies \lambda_1=...=\lambda_{n-1}=0$ ?
It makes sense of course but is it enough for the proof?
Thanks for your help