Snails and Sums 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.

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**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.)_