We show that if \(\gamma : S^1 \rightarrow \mathbb {R}^2\) is a Jordan curve which is close to a \(C^2\) Jordan curve \(\beta : S^1 \rightarrow \mathbb {R}^2\) , then \(\gamma \) contains an inscribed square of positive sidelength. In particular, if \(\kappa > 0\) is the maximum unsigned curvature of \(\gamma \) and there is a continuous function f from the image of \(\beta \) to the image of \(\gamma \) so that \(\begin{aligned} |f(\beta (s)) - \gamma (s)| < \frac{1}{10\kappa } \end{aligned}\) for every \(s \in S^1\) and \(f \circ \beta \) is homotopic to \(\gamma \) in the image of \(\gamma \) , then \(\beta \) contains a square of positive sidelength.
