At the beginning of a $10\,\mathrm m$ long rubber band sits a snail. Every day it crawls one meter ahead. Every night, when the snail is resting, an evil man stretches the tape evenly by $10\,\mathrm m$ out. (In the morning of the second Day the band is $20\,\mathrm m$ long and the snail has made $2\,\mathrm m$.) Will the snail ever reach the end of the tape?
My attempt:
If this continues forever, we have $d_{\text{Snail}}=\sum_{n=0}^{\infty}{1}$ and $d_{\text{Tape}}=\sum_{n=0}^{\infty}{10}$. They are both equal to $\infty$ since $a\cdot \infty = \infty$. But how does that prove it?
Instead of measuring the snail's (or snake's?) progress in meters, measure it in fractions of the entire rubber band, since that is a measure that stays the same during the nightly stretching.
Then on day $n$ it makes a progress of $1/10n$ rubber bands. So you want find out whether there's a $k$ such that $$ \sum_{n=1}^{k} \frac1{10n} \ge 1 $$ and we know this to be the case since the harmonic series diverges.
A more vivid argument: Start by painting 20 dots on the rubber band at equal intervals. The rubber between two dots stretches by 50 cm every night, but since the snail crawls at 100 cm a day, each morning it will be at least 50 cm closer to the next dot than it was the previous morning (unless it passed a dot yesterday). So it will inevitably reach the next dot. And all it needs to do to get all the way is to "reach the next dot" 20 times in a row.
(Any fencepost errors in this argument are left for the reader to think around.)