One way to prove that $\displaystyle P \longrightarrow Q$ is to show that $\displaystyle - Q \longrightarrow -P$, which is what I did. If $\displaystyle u,v,w$ are linearly dependent then $\displaystyle S$ must be linearly dependent. Thus if $\displaystyle S$ is linearly independent we must necessarily have that $\displaystyle u,v,w$ are also linearly independent.
The proof is given. Which part don't you understand?
if $\displaystyle u,v$ and $\displaystyle w$ were dependent,$\displaystyle \text{dim}_{\mathbb{R}}Span(u,v,w)\leq 2$,therefore $\displaystyle u,v$ and $\displaystyle w$ will be in $\displaystyle Span(u,v,w) $ with a dimension less than 2,hence they can't be independent.
is that what you're trying to say
P.S: i like your proof i just can't see through it.