The set of all possible submarine routes is countable (velocity vector + position at time 0). Let's assume we have an enumeration of that set. Now it's easy to bomb them all. At time 1 we bomb the point where submarine 1 is at time 1, at time 2 we bomb the point where submarine 2 is at time 2, and so on.

Here's a link to the proof I typed up for my problem set:

https://www.sharelatex.com/project/534230c0234f079f3ce526fa?...

Theorem 1.10 states "If S andT are countable sets, the set S ×T ={(s,t) : s∈S,t∈T} is countable." and is proven using a diagonal argument.