Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded/Proof 1

Proof
From Convergent Sequence in Topological Vector Space is Cauchy, $\sequence {x_n}_{n \in \N}$ is a Cauchy sequence.

From Image of Cauchy Sequence in Topological Vector Space is von Neumann-Bounded, $E$ is von Neumann-bounded.