A common question in analysis is: Let $I$ be an open interval in $\mathbb{R}$. Show that a continuous function $f:I \to \mathbb{R}$ is uniformly continuous iff for all sequences $(x_n), (y_n) \subset I$ such that $\vert x_n - y_n \vert \to 0$ then $\vert f(x_n) - f(y_n) \vert \to 0$.
The standard way to prove the converse direction is via the contrapositive, see for example: Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with $d(x_n, y_n) \to 0$.
How would I prove it directly? ie, without the need for contrapositive.