Here is a proof I found. 
[![enter image description here](https://i.stack.imgur.com/cfeZR.png)](https://i.stack.imgur.com/cfeZR.png)
[![enter image description here](https://i.stack.imgur.com/FLykT.png)](https://i.stack.imgur.com/FLykT.png)

To prove a sequence of points converges at a point $x$, I can show that for any neighborhood $U$ of $x$, there exists an integer $N$ such that for any $n \geq N$, $x\_n \in U$. I don't understand why did he prove "for any subbasis set S for the product topology..."? [This proof](https://math.stackexchange.com/questions/524481/convergence-in-product-topology?rq=1) makes more sense to me.