I'm looking for a proof verification of a solution of seemingly easy exercise. I ask because I often mess these geometrical proofs up.
Exercise. Let $\gamma :[0,1]\to \mathbb R^2$ be a smooth parametrized curve. Prove its image cannot contain the unit square.
Attempt. By definition the arclength of a parametrized curve is the supremum of lengths polygonal approximations of its image over all partitions of its domain. If a parametrized curve is $C^1$ it has finite arclength. We shall find a divergent sequence of lengths of polygonal approximations of $\gamma$, proving the supremum is infinite and hence contradicting the assumption $\gamma$ is $C^1$.
Divide $[0,1]^2$ into $n^2$ congruent squares in the obvious way. $\gamma$ must pass through every corner of each small square in some order. In particular $\gamma$ must pass through the bottom left corner of each square in some order. Take the polygonal approximation at these corners. The distance between bottom left corners is at least $\frac 1n$ but there are $n^2$ to traverse, so the length of the polygonal approximation is at least $n$. Repeating for each $n$ we obtain a divergent sequence of lengths of polygonal approximations.