Trying to prove the following.
If $\left(x_{\lambda}\right)_{\lambda\in\Lambda}$ is a net in $\prod X_{\alpha}$ having $x$ as a cluster point (limit point) then for each $\alpha$, $\left(\pi_{\alpha}\left(x_{\lambda}\right)\right)_{\lambda\in\Lambda}$ has $\pi_{\alpha}(x)$ for a cluster point.
In other words, I am trying to show that (I think?) for each $\alpha$,
$$\forall U\text{ open}:\pi_{\alpha}(x)\in U, \forall\mu\in\Lambda, \exists\lambda\in\Lambda:\pi_{\alpha}\left(x_{\lambda}\right)\in U.$$
Intuitivly, this makes sense as an extension of sequence, however, when actually trying to formulate a proof, I get immediately stuck. A proof of it or any remarks would be helpful.
Not quite: you need to show that if $U$ is an open nbhd of $\pi_\alpha(x)$, then for each $\mu\in\Lambda$ there is a $\lambda\in\Lambda$ such that $\color{red}{\mu\le\lambda}$ and $\pi_\alpha(x_\lambda)\in U$. This follows directly from the continuity of $\pi_\alpha$: $\pi_\alpha^{-1}[U]$ is an open nbhd of $x$ in the product, so for each $\mu\in\Lambda$ there is a $\lambda\in\Lambda$ such that $\mu\le\lambda$ and $x_\lambda\in\pi^{-1}[U]$, and hence $\pi_\alpha(x_\lambda)\in U$.