We will prove the contrapositive (Proof Technique CP): If $T$ is linearly dependent, then $S$ is linearly dependent. This might be an interesting statement in its own right.
Write $S=\set{\vectorlist{v}{m}}$ and without loss of generality we can assume that the subset $T$ is the first $t$ vectors of $S$, $t\leq m$, so $T=\set{\vectorlist{v}{t}}$. Since $T$ is linearly dependent, by Definition LI there are scalars, not all zero, $\scalarlist{a}{t}$, so that
\begin{align*}
\zerovector
&= \lincombo{a}{v}{t}\\
&= \lincombo{a}{v}{t}+\zerovector+\zerovector+\dots+\zerovector\\
&= \lincombo{a}{v}{t}+0\vect{v}_{t+1}+0\vect{v}_{t+2}+\dots+0\vect{v}_{m}
\end{align*}
which is a nontrivial relation of linear dependence (Definition RLD) on the set $S$, so we can say $S$ is linearly dependent.