So my suggestion was:
Assume: Let $S = \{u_1, u_2, ....u_r\}$ be a set of vectors in $\mathbb{R}^n$ . If $r > n$ then the vectors $u_1, u_2 \ldots u_r$ must be linearly dependent.
If one writes the linear system corresponding to $c_1u_1 + c_2u_2 + ... + c_ru_r$ , one will have a homogeneous system of $n$ equations in $r$ unknowns. We know that such a system has infinitely many solutions. Thus in $\mathbb{R}^n$ , a set which is linearly independent cannot contain more than $n$ vectors.