Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded/Proof 2

Proof
From Union of Image of Convergent Sequence and Limit in Hausdorff Space is Compact, $E \cup \set x$ is compact.

From Compact Subspace of Topological Vector Space is von Neumann-Bounded, $E \cup \set x$ is von Neumann bounded.

From Subset of von Neumann-Bounded Set is von Neumann-Bounded, $E$ is von Neumann bounded.