Everything is in the title !
Ribes & Zaleski propose (cor 1.1.8 of"Profinite Groups") as an exercise to prove $\overline Y=\varprojlim \pi_i(Y)$ where $Y\subset\varprojlim E_i$ is a subset of a projective limit of compact-Hausdorff sets $E_i$ and $\pi_i$ are the projections.
I just know $Y\subset\varprojlim \pi_i(Y)\subset\overline Y$ and that $Y$ is dense in $\varprojlim \pi_i(Y)$.
The authors sugjest to prove that $\varprojlim \pi_i(Y)$ is closed but I don't see how.
If I take a point $x=(x_i)\notin\varprojlim \pi_i(Y)$ i can just say that it exists a $i\in I$ such that $x_i\notin\pi_i(Y)$ and so for $i\leq j$, $x_j\notin\pi_j(Y)$ an so what ?
Have someone an idea ?