Let $p \geq 1$, and let $\mathbf{X}$ and $\mathbf{X}^{(n)}$ for $n \geq 1$ be (geometric) $p$-rough paths on $[0,1]$ — that is, paths enhanced with iterated integrals up to order $\lfloor p \rfloor$, viewed as elements of the space of $p$-rough paths. Suppose $\mathbf{X}^{(n)} \to \mathbf{X}$ in the $p$-variation rough path metric, and let $f \in C^{\lfloor p \rfloor + 1}_b(\mathbb{R}^e; \mathcal{L}(\mathbb{R}^d, \mathbb{R}^e))$. Then the solutions $y^{(n)} \in C^{p\text{-var}}([0,1]; \mathbb{R}^e)$ of the rough differential equations $dy^{(n)}_t = f(y^{(n)}_t)\, d\mathbf{X}^{(n)}_t$ converge uniformly on $[0,1]$ to the solution $y \in C^{p\text{-var}}([0,1]; \mathbb{R}^e)$ of $dy_t = f(y_t)\, d\mathbf{X}_t$.