Dilation of Closed Set in Topological Vector Space is Closed Set/Proof 1

Proof
We aim to show that $X \setminus \paren {\lambda F}$ is open.

Since $F$ is closed, $X \setminus F$ is open.

It follows from Dilation of Open Set in Topological Vector Space is Open that $\lambda \paren {X \setminus F}$ is open.

From Dilation of Complement of Set in Vector Space, we have:


 * $X \setminus \paren {\lambda F} = \lambda \paren {X \setminus F}$.

Since we have established that $\lambda \paren {X \setminus F}$ is open, it follows that $X \setminus \paren {\lambda F}$ is open.

So $\lambda F$ is closed.