n+1 vectors in $\mathbb{R}^n$ cannot be linearly independent

n+1 vectors in $\mathbb{R}^n$ cannot be linearly independent

I was looking for a short snazzy proof on the following statement:
n+1 vectors in $\mathbb{R}^n$ cannot be linearly independent
A student of mine asked this today morning and I couldn't come up with a
proof solely from the definition of linear independence.
From a higher level perspective, I explained that if I put the vectors in
a matrix then if the only null space entry is the zero vector, then the
vectors are independent but since we have one extra column than row and
that the row and column rank are equal, there is no way we can have $n+1$
as the rank of the matrix and hence from Rank-Nullity theorem, the
dimension of the Nullspace is at least one which implies that there is a
combination of the vectors where not all the scalar multiples in the
definition are 0 but yet we get a zero as the linear combination. The
student hasn't completely learnt the fundamental subspaces yet so I am not
sure he grasped what I was saying.
Is there a cleaner proof?
EDIT: I am stunned how many beautiful answers I got with so much diversity.