Lemma 4

Let E be a Fréchet space and A a compact subset of E. There exists a compact convex balanced subset L of E containing A such that the topologies induced on A by those of E and of $E_L$ coincide.

The set A is contained in the closed convex balanced hull of the set of points of a sequence $(x_n)_{n\in\mathbf{N}}$ of elements of E converging to 0 (EVT, IV, p. 24, Cor. 1). Let $(y_n)_{n\in\mathbf{N}}$ and $(\lambda_n)_{n\in\mathbf{N}}$ be sequences satisfying the conclusions of Lemma 3. The closed convex balanced hull L of the points of the sequence $(y_n)$ contains A and is a compact subset of E (EVT, III, p. 8). Hence $E_L$ is a Banach space (EVT, III, p. 8, Cor.). In this Banach space, the norm of $x_n$ is majorized by $\lambda_n$, therefore the sequence $(x_n)$ tends to 0. The closed convex balanced hull $\widetilde{A}$ of the sequence $(x_n)$ in $E_L$ is a compact subset of $E_L$ (EVT, III, p. 8). Since L is a bounded subset of E, the canonical injection of $E_L$ into E is continuous. The topologies induced on $\widetilde{A}$ by those of $E_L$ and of E coincide, and $\widetilde{A}$ is a compact, hence closed, subset of E. Since the set $\widetilde{A}$ is convex, balanced, and contains the sequence $(x_n)$, it contains A. This completes the proof.