I was reading Milne's notes on class field theory. On page 22, while giving an example to the fact that every subgroup of $K^{\times}$, where K is a local field of characteristic $p\neq 0$, is not open, he makes the following remark:
Choose a subgroup of finite index in $\underset{\mathbb N}\prod \mathbb F_p$, which contains $\underset{\mathbb N}\bigoplus \mathbb F_p$.
I am not able to figure out why such a subgroup exists? Please help.