We want to prove that any set of $t+1$ or more vectors from $V$ is linearly dependent. So we will begin with a totally arbitrary set of vectors from $V$, $R=\set{\vectorlist{u}{m}}$, where $m>t$. We will now construct a nontrivial relation of linear dependence on $R$.
Each vector $\vectorlist{u}{m}$ can be written as a linear combination of the vectors $\vectorlist{v}{t}$ since $S$ is a spanning set of $V$. This means there exist scalars $a_{ij}$, $1\leq i\leq t$, $1\leq j\leq m$, so that \begin{align*} \vect{u}_1&=a_{11}\vect{v}_1+a_{21}\vect{v}_2+a_{31}\vect{v}_3+\cdots+a_{t1}\vect{v}_t\\ \vect{u}_2&=a_{12}\vect{v}_1+a_{22}\vect{v}_2+a_{32}\vect{v}_3+\cdots+a_{t2}\vect{v}_t\\ \vect{u}_3&=a_{13}\vect{v}_1+a_{23}\vect{v}_2+a_{33}\vect{v}_3+\cdots+a_{t3}\vect{v}_t\\ &\quad\quad\vdots\\ \vect{u}_m&=a_{1m}\vect{v}_1+a_{2m}\vect{v}_2+a_{3m}\vect{v}_3+\cdots+a_{tm}\vect{v}_t \end{align*}
Now we form, unmotivated, the homogeneous system of $t$ equations in the $m$ variables, $x_1,\,x_2,\,x_3,\,\ldots,\,x_m$, where the coefficients are the just-discovered scalars $a_{ij}$, \begin{align*} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1m}x_m&=0\\ a_{21}x_1+a_{22}x_2+a_{23}x_3+\cdots+a_{2m}x_m&=0\\ a_{31}x_1+a_{32}x_2+a_{33}x_3+\cdots+a_{3m}x_m&=0\\ \vdots\quad\quad&\\ a_{t1}x_1+a_{t2}x_2+a_{t3}x_3+\cdots+a_{tm}x_m&=0\\ \end{align*}
This is a homogeneous system with more variables than equations (our hypothesis is expressed as $m>t$), so by Theorem HMVEI there are infinitely many solutions. Choose a nontrivial solution and denote it by $x_1=c_1,\,x_2=c_2,\,x_3=c_3,\,\ldots,\,x_m=c_m$. As a solution to the homogeneous system, we then have \begin{align*} a_{11}c_1+a_{12}c_2+a_{13}c_3+\cdots+a_{1m}c_m&=0\\ a_{21}c_1+a_{22}c_2+a_{23}c_3+\cdots+a_{2m}c_m&=0\\ a_{31}c_1+a_{32}c_2+a_{33}c_3+\cdots+a_{3m}c_m&=0\\ \vdots\quad\quad&\\ a_{t1}c_1+a_{t2}c_2+a_{t3}c_3+\cdots+a_{tm}c_m&=0\\ \end{align*}
As a collection of nontrivial scalars, $c_1,\,c_2,\,c_3,\,\dots,\,c_m$ will provide the nontrivial relation of linear dependence we desire, \begin{align*} &\lincombo{c}{u}{m}\\ &=c_{1}\left(a_{11}\vect{v}_1+a_{21}\vect{v}_2+a_{31}\vect{v}_3+\cdots+a_{t1}\vect{v}_t\right) &&\knowl{./knowls/definition.SSVS.knowl}{\text{Definition SSVS}}\\ &\quad\quad+c_{2}\left(a_{12}\vect{v}_1+a_{22}\vect{v}_2+a_{32}\vect{v}_3+\cdots+a_{t2}\vect{v}_t\right)\\ &\quad\quad+c_{3}\left(a_{13}\vect{v}_1+a_{23}\vect{v}_2+a_{33}\vect{v}_3+\cdots+a_{t3}\vect{v}_t\right)\\ &\quad\quad\quad\quad\vdots\\ &\quad\quad+c_{m}\left(a_{1m}\vect{v}_1+a_{2m}\vect{v}_2+a_{3m}\vect{v}_3+\cdots+a_{tm}\vect{v}_t\right)\\ &=c_{1}a_{11}\vect{v}_1+c_{1}a_{21}\vect{v}_2+c_{1}a_{31}\vect{v}_3+\cdots+c_{1}a_{t1}\vect{v}_t &&\knowl{./knowls/property.DVA.knowl}{\text{Property DVA}}\\ &\quad\quad+c_{2}a_{12}\vect{v}_1+c_{2}a_{22}\vect{v}_2+c_{2}a_{32}\vect{v}_3+\cdots+c_{2}a_{t2}\vect{v}_t\\ &\quad\quad+c_{3}a_{13}\vect{v}_1+c_{3}a_{23}\vect{v}_2+c_{3}a_{33}\vect{v}_3+\cdots+c_{3}a_{t3}\vect{v}_t\\ &\quad\quad\quad\quad\vdots\\ &\quad\quad+c_{m}a_{1m}\vect{v}_1+c_{m}a_{2m}\vect{v}_2+c_{m}a_{3m}\vect{v}_3+\cdots+c_{m}a_{tm}\vect{v}_t\\ &=\left(c_{1}a_{11}+c_{2}a_{12}+c_{3}a_{13}+\cdots+c_{m}a_{1m}\right)\vect{v}_1 &&\knowl{./knowls/property.DSA.knowl}{\text{Property DSA}}\\ &\quad\quad+\left(c_{1}a_{21}+c_{2}a_{22}+c_{3}a_{23}+\cdots+c_{m}a_{2m}\right)\vect{v}_2\\ &\quad\quad+\left(c_{1}a_{31}+c_{2}a_{32}+c_{3}a_{33}+\cdots+c_{m}a_{3m}\right)\vect{v}_3\\ &\quad\quad\quad\quad\vdots\\ &\quad\quad+\left(c_{1}a_{t1}+c_{2}a_{t2}+c_{3}a_{t3}+\cdots+c_{m}a_{tm}\right)\vect{v}_t\\ &=\left(a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}+\cdots+a_{1m}c_{m}\right)\vect{v}_1 &&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &\quad\quad+\left(a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}+\cdots+a_{2m}c_{m}\right)\vect{v}_2\\ &\quad\quad+\left(a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}+\cdots+a_{3m}c_{m}\right)\vect{v}_3\\ &\quad\quad\quad\quad\vdots\\ &\quad\quad+\left(a_{t1}c_{1}+a_{t2}c_{2}+a_{t3}c_{3}+\cdots+a_{tm}c_{m}\right)\vect{v}_t\\ &=0\vect{v}_1+0\vect{v}_2+0\vect{v}_3+\cdots+0\vect{v}_t &&\text{$c_j$ as solution}\\ &=\zerovector+\zerovector+\zerovector+\cdots+\zerovector &&\knowl{./knowls/theorem.ZSSM.knowl}{\text{Theorem ZSSM}}\\ &=\zerovector &&\knowl{./knowls/property.Z.knowl}{\text{Property Z}} \end{align*}
That does it. $R$ has been undeniably shown to be a linearly dependent set.
The proof just given has some monstrous expressions in it, mostly owing to the double subscripts present. Now is a great opportunity to show the value of a more compact notation. We will rewrite the key steps of the previous proof using summation notation, resulting in a more economical presentation, and even greater insight into the key aspects of the proof. So here is an alternate proof — study it carefully.
Alternate Proof: We want to prove that any set of $t+1$ or more vectors from $V$ is linearly dependent. So we will begin with a totally arbitrary set of vectors from $V$, $R=\setparts{\vect{u}_j}{1\leq j\leq m}$, where $m>t$. We will now construct a nontrivial relation of linear dependence on $R$.
Each vector $\vect{u_j}$, $1\leq j\leq m$ can be written as a linear combination of $\vect{v}_i$, $1\leq i\leq t$ since $S$ is a spanning set of $V$. This means there are scalars $a_{ij}$, $1\leq i\leq t$, $1\leq j\leq m$, so that \begin{align*} \vect{u}_j&=\sum_{i=1}^{t}a_{ij}\vect{v}_i&&1\leq j\leq m \end{align*}
Now we form, unmotivated, the homogeneous system of $t$ equations in the $m$ variables, $x_j$, $1\leq j\leq m$, where the coefficients are the just-discovered scalars $a_{ij}$, \begin{align*} \sum_{j=1}^{m}a_{ij}x_j=0&&1\leq i\leq t \end{align*}
This is a homogeneous system with more variables than equations (our hypothesis is expressed as $m>t$), so by Theorem HMVEI there are infinitely many solutions. Choose one of these solutions that is not trivial and denote it by $x_j=c_j$, $1\leq j\leq m$. As a solution to the homogeneous system, we then have $\sum_{j=1}^{m}a_{ij}c_{j}=0$ for $1\leq i\leq t$. As a collection of nontrivial scalars, $c_j$, $1\leq j\leq m$, will provide the nontrivial relation of linear dependence we desire, \begin{align*} \sum_{j=1}^{m}c_{j}\vect{u}_j &=\sum_{j=1}^{m}c_{j}\left(\sum_{i=1}^{t}a_{ij}\vect{v}_i\right) &&\knowl{./knowls/definition.SSVS.knowl}{\text{Definition SSVS}}\\ &=\sum_{j=1}^{m}\sum_{i=1}^{t}c_{j}a_{ij}\vect{v}_i &&\knowl{./knowls/property.DVA.knowl}{\text{Property DVA}}\\ &=\sum_{i=1}^{t}\sum_{j=1}^{m}c_{j}a_{ij}\vect{v}_i &&\knowl{./knowls/property.C.knowl}{\text{Property C}}\\ &=\sum_{i=1}^{t}\sum_{j=1}^{m}a_{ij}c_{j}\vect{v}_i &&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &=\sum_{i=1}^{t}\left(\sum_{j=1}^{m}a_{ij}c_{j}\right)\vect{v}_i &&\knowl{./knowls/property.DSA.knowl}{\text{Property DSA}}\\ &=\sum_{i=1}^{t}0\vect{v}_i &&\text{$c_j$ as solution}\\ &=\sum_{i=1}^{t}\zerovector &&\knowl{./knowls/theorem.ZSSM.knowl}{\text{Theorem ZSSM}}\\ &=\zerovector &&\knowl{./knowls/property.Z.knowl}{\text{Property Z}} \end{align*}
That does it. $R$ has been undeniably shown to be a linearly dependent set.