The exercise had one hint: Use an algebraic basis of $X$ and Problem 22. And the Problem 22: Let $X$ be a metrizable TVS and let $(x_n)$ be a sequence of elements of $X$. Show that there exists a sequence of strictly positive numbers $(λ_n)$ such that $λ_n x_n \rightarrow 0_X$.
And the Problem 22 had one hint too: Let ρ be a metric generating the topology on X. Show and then use that $\displaystyle\lim_{t\rightarrow 0+} \rho(0_X, tx) = 0$ for each $x \in X$.