Assume $(X,\|\cdot\|_X)$ is a normed space and if $Y$ is a dense linear subspace of $X$,
How to prove that for each $x \in X$ there exists a sequence $(y_j)_j \subset Y$ such that $$\sum_{j=1}^\infty y_j = x$$ in $X$ and $$\sum_{j=1}^\infty\|\mathbf{y}_j\|_X \leq 2\|x\|_X$$
Take $y_1$ within $||x||/2$ of $x$, with $||y_1||\leq ||x||$. Next $x-y_1$ has norm $\leq ||x||/2$, so take $y_2$ within $||x||/4$ of $x-y_1$, with $||y_2||\le ||x||/2$. So $x-y_1-y_2$ has norm $\leq ||x||/4$, and $||y_2||\le \frac{1}{2}||x||$.
Keep going, so we get a sequence of $y_i$ such that $||y_i||\le 2^{-i+1}||x||$, while $||x-\sum_{i=1}^ny_i||\leq 2^{-n}||x||$. So this is the sequence you were searching for. Indeed, the sum is clearly Cauchy hence the infinite sum exists and is equal to $x$, and the infinite sum has the wanted bound:$$\sum_{i=1}^\infty ||y_i||\le \sum_{i=1}^\infty 2^{-i+1}||x||=2\cdot ||x||$$