Limit points of the following set
Solution 1:
The set of limit points $\{[\pi_1, \pi_2, \pi_3]^{T}: \pi_i \geq 0, \sum \pi_i=1\}$.
It is clear that limit points have to lie in this set. Let $[\pi_1,\pi_2,\pi_3]^{T}$ be any point in this set. There exists $i$ such that $\pi_i >0$. Take $j \neq i$. Now consider the sequence $[\pi_{1n}',\pi_{2n}',\pi_{3n}']^{T}$ where $\pi_{in}'=\pi_i-\frac 1 n, \pi_{jn}'=\pi_j+\frac 1 n$ and $\pi_{kn}'=\pi_k$ for $k \notin \{i,j\}$. This sequence converges to $[\pi_1,\pi_2,\pi_3]^{T}$