TITLE: A problem related to the submarine puzzle
QUESTION [7 upvotes]: Submarine puzzle: 

A submarine is located at an integer somewhere along the number line.
  It is moving at some integral velocity (an integral number of units
  per second). Every second you may drop a bomb which will destroy the
  submarine if the submarine is at that location.
Can you be guaranteed of destroying the submarine? If so, what
  strategy would you use?

(from http://math-fail.com/2011/01/submarine-puzzle.html)
And the answer is to enumerate all $(a,b)$ pairs, where $a, b$ are integers and the location of the submarine at time $t$ equals to $at+b$.
My question is, if now $a, b$ can be any real numbers, and the bomb can now destroy the submarine in a region of length 1,
i.e. if at time $t$ the bomb is dropped at $x_t$ and $x_t - 0.5 \le at+b \le x_t+0.5$,
the submarine will be destroyed.
Then can we be guaranteed to destroy the submarine?
Some thinking:
When $b$ is given, the mission can be done by enumerating $a$ in all rational number (since rational number is dense).
When $a$ is restricted to an integer, the mission can be done by the same method as above.
For general case, I think it is not possible to destroy the submarine, but I cannot prove it.

REPLY [3 votes]: Yes, the submarine can be destroyed. For simplicity assume the first bomb is $n=0$, and with the $n$-th bomb you wipe out a stripe in the $a,b$ plane of horizontal width 1 and slope $-1/n$. Subdivide the plane into unit squares, and enumerate the squares so that the plane is covered by countably many of them. If, starting at arbitrary bomb $n$, we can cover a unit square in finitely many bombs, we are able to cover the whole plane in countably many bombs and are guaranteed to destroy he submarine.
Pick the position of bomb $n$ so that the stripe covers the bottom edge of the square, which means it covers the bottom $1/n$ of the left edge. Pick the next bomb so it covers the bottom $1/(n+1)$ of the right edge. That stripe has a shallower slope than the previous: they overlap. The next bomb covers the next $1/(n+2)$ of the right edge, the following the next $1/(n+3)$ of the right edge, etc. It will require finitely many bombs to completely cover the unit square.
Since this is true for any unit square, starting with any $n$, we can completely cover the $a,b$ plane, hence destroy the submarine no matter where it started or its speed.
Apologies for brevity, I will expand my explanation from someplace other than my phone.