Let $\varprojlim {A_i} = P$ with $\mu_{ji} : A_j \to A_i$ be surjective. Show $\pi_i:P \to A_i$ is surjective for all $i\in\mathbb{Z}^+$.
Let $a_i \in A_i$. Then for all $j > i$, the set $\mu_{ji}^{-1}(a_i)$ is non empty.
Take $a = (\mu_{i1}(a_i),\mu_{i2}(a_i), ..., a_i, a_{i+1}, ... )$ where $a_j \in \mu_{ji}^{-1}(a_i)$ for $j>i$
Clearly $a\in P$ since $\mu_{ji}(a_j) = a_i$ and $\pi_i(a) = a_i$.
Is my attempt correct?