Suppose you were walking from the top left corner of a large square $S$ with side length $1$ to the bottom right. To do so, you go to from the top left corner to the center of square $S$ in a right then down matter, then go from the center to the bottom right corner in a down then right path. The total length of this path is 2. Now, let $S_1$ be the square with opposing corners being the top left corner and the center of $S$, and $S_2$ be the square with opposing corners being the center and the bottom right corner, and do the same game with $S_1$ and $S_2$. (This is the second picture.) Now let $S_{1, 1}$ and $S_{1, 2}$ be the two squares made in a similar fashion to $S_1$ and $S_2$, but with the role of $S$ being played by $S_1$, and let $S_{2, 1}$ and $S_{2, 2}$ be made similarly, and again play this same game (This is the third picture.) Note that the lengths of the paths are always $2$, since we never add or subtract any excess length. This means that the sequence of lengths has limit $2$. However, as we repeat this process recursively, the total path approaches the diagonal of the square, which has length $\sqrt{2}$, as the maximum vertical distance between the diagonal and the path approaches $0$. This means that $2=\sqrt{2}$.