Let $E$ be a (not necessarily Hausdorff) real TVS. I'm trying to solve exercise 13 in these notes by professor Gabriel Nagy
Let $(x_d)_{d\in D}$ be a net in $E$ such that $x_d \to 0_E$. Let $(\lambda_d)_{d\in D}$ be a net in $\mathbb R$ such that $M :=\sup_{d\in D} |\lambda_d| <\infty$. Then $\lambda_d x_d \to 0_E$.
There are possibly subtle mistakes that I could not recognize in below attempt. Could you please have a check on it?
Let $U$ be an open neighborhood (nbh) of $0_E$.
Let $T:\mathbb R \times E \to E, (t, x) \mapsto tx$ be the multiplication map. Then $T$ is continuous. Then there exist an open nbh $V_1$ pf $0_\mathbb R$ and an open nbh $U_1$ of $0_E$ such that $T(V_1 \times U_1 ) \subset U$.
There is $t_1>0$ such that $I :=(-t_1, t_1) \subset V_1$. Let $I_1 := (-2M, 2M)$ and $U_2 := \frac{t_1}{2M} U_1$. Then $U_2$ is an open nbh of $0_E$, and $$ T(I_1 \times U_2) \subset U. \tag{$*$} $$
It follows from $x_d \to 0_E$ that there is $\bar d \in D$ such that $x_d \in U_2$ for all $d \in D$ with $\bar d < d$. By $(*)$, $\lambda_d x_d \in U$ for all $d \in D$ with $\bar d < d$. This completes the proof.