Source

I got this problem from Rustan Leino, who got it from Jay Misra.

I solved it and wrote up my solution.

Problem

A rubber band (well, a rubber string, really) is 10 meters long. There's a worm that starts at one end and crawls toward the other end, at a speed of 1 meter per hour. After each hour that passes, the rubber string is stretched so as to become 1 meter longer than it just was. Will the worm ever reach the other end of the string?

Solution
Reveal

Yes, and here's why. Let $f$ represent the fraction of the string that the worm has traveled. At the beginning of the $k$th hour, the string will have length $k+9$ meters and the worm will travel 1 meter, thus increasing $f$ by $1/(k+9)$. The string will then stretch, but this doesn't change $f$. For this reason, after $n$ hours, $f$ will be $\sum_{k=1}^n 1/(k+9)$. This is a suffix of the harmonic series, which is known to diverge. So, eventually $f$ will exceed any arbitrary amount. In particular, it will eventually exceed 1, which is when the worm will be done. (Some calculation reveals that this will happen during the 17th hour.)