We define mutually unbiased bases (MUBs) as
Let $\mathcal{B}_1, ..., \mathcal{B}_k$ be $k$ orthonormal bases for $\mathbb{C}^n$. We call these bases mutually unbiased if they obey $$ \forall v \in \mathcal{B}_i, w \in \mathcal{B}_j, i\neq j \qquad |\langle v,w \rangle |^2 = \frac{1}{n}.$$
Multiple articles state that the maximum number of MUBs is less than or equal to $n+1$, but do not prove this. I do not see however why this is so easy to prove?
EDIT: I have now found a proof in the article 'A New Proof for the Existence of Mutually Unbiased Bases' by Somshubhro Bandyopadhyay, P. Oscar Boykin, Vwani Roychowdhury, and Farrokh Vatan. However, their proof is quite elaborate and definitely not as easy as many authors make it out to be? It does answer my question, but the question that remains is: Is there a simpler/shorter approach?