  We show that if $\gamma$ is a Jordan curve in $\mathbb{R}^2$ which is close
to a $C^2$ Jordan curve $\beta$ in $\mathbb{R}^2$, then $\gamma$ contains an
inscribed square. In particular, if $\kappa > 0$ is the maximum unsigned
curvature of $\beta$ and there is a map $f$ from the image of $\gamma$ to the
image of $\beta$ with $||f(x) - x|| < \frac{1}{10 \kappa}$ and $f \circ \gamma$
having winding number $1$, then $\gamma$ has an inscribed square of positive
sidelength.
