> Here is a tedious answer.

It is straightforward to see that for $n=1$ there at at most $2n$ vectors satisfying the conditions.

Suppose it is true for $1,..,n$.
Further, suppose we have $a_1,...,a_{2(n+1)+1}$ points in $\mathbb{R}^{n+1}$ that satisfy the conditions. This will lead to a contradiction.

Let $b_j = \operatorname{proj}_{ \operatorname{sp} \{ a_1 \}^\bot } a_j$, then the $b_j$ are all in $a_1^\bot$ (which is essentially $\mathbb{R}^{n}$). It is straightforward to verify that $\langle b_i, b_j \rangle \le 0$ for $i \neq j$, hence the inductive hypothesis shows that $b_2,...,b…

Let $P$ be an $N \times N$ transition matrix which is irreducible and aperiodic. Perron-Frobenius theorem states that $1$ is the unique largest eigenvalue of $P$ and for all other eigenvalues $\lambda$ of $P$ we
have $|\lambda|<1$. Let $\pi=(\pi_1, \pi_2, \ldots, \pi_N)$ be the left eigenvector of $P$ corresponding to $1$ such that $\sum_{i=1}^N \pi_i=1$. Use Perron-Frobenius theorem and linear algebra techniques (e.g. Jordan Canonical Form) to prove that as $t \to \infty$, $(P^t)_{ij} \to \pi_j \text{ for all } i,j\, .$