Denseness of $\ell_1$ in a bigger normed vector space of sequences

Question

Let $(x_n)_{n\ge 1}$ be a given real sequence such that $$ \sum_{n\le N}|x_n|=o(N) \,\,\,\,\text{ as }N\to \infty. $$ Is it true that, for each $\varepsilon>0$, there exists a real sequence $(y_n)_{n\ge 1}$ such that $\sum_{n=1}^\infty|y_n|<\infty$ and $$ \forall N\ge 1,\,\,\,\,\sum_{n\le N}|x_n-y_n| \le N\varepsilon\,\,? $$

Answer 1

Let $S_N = \sum_{n \le N} |x_n|$. Given $\epsilon > 0$, since $S_N = o(N)$ there exists $N_0$ such that $S_N \le \epsilon N$ for all $N > N_0$. Let $$y_n = \begin{cases} x_n, & n \le N_0 \\ 0, & n > N_0 \end{cases}.$$ Clearly $\sum |y_n| < \infty$. For $N \le N_0$ we have $\sum_{n \le N} |x_n - y_n| = 0 \le \epsilon N$, and for $N > N_0$ we have $\sum_{n\le N} |x_n - y_n| = \sum_{n=N_0+1}^N |x_n| \le S_N \le \epsilon N$.