Proof. (2) $\Rightarrow$ (3): Let $B \subset E$ be a bounded set, $\seq{x_n}\subset E$, and $\seq{\lambda_n}\subset K$ such that $\lambda_{n} \to 0$ as $n \to \infty$, then $\lambda_{n} x_{n} \to 0$ as $n \to \infty$. By sequential continuity of $T$, $T(\lambda_{n} x_{n}) \to 0$ as $n \to \infty$ as well. Thus $T(B)$ is also bounded.

(3) $\Rightarrow$ (1): Let $\rho: F \to [0, \infty)$ be a continuous seminorm, then $\rho \circ T$ is a seminorm on $E$ that is bounded on bounded sets. Since $E$ is bornological, $\rho \circ T$ is continuous. Therefore $T$ is continuous by Proposition 11.2.1.$\square$