A convergent sequence in $\ell_1$ that is $1/2$-distant to the standard basis

Question

I was thinking about the following, it should be easy, but somehow I am unable to find an example. Let $(e_k)$ be the unit vector basis of $\ell_1$. Can one give me an example of a convergent sequence $(x_k)$ in $\ell_1$ such that $\|e_k-x_k\|\leqslant 1/2$ for all $k$?

Answer 1

Write $x_k = (x_k^1,x_k^2,x_k^3,\ldots)$. Since $$|x_k^k - 1| = |x_k^k - e_k^k| \le \|x_k - e_k\| \le \frac 12$$ for all $k$ you have $x_k^k \ge \frac 12$ for all $k$.

Suppose that $x_k \to x$ in $\ell_1$. Then there exists an index $N$ with the property that $k \ge N$ implies $\|x_k - x\| < \dfrac 14$. In particular, $$k \ge N \implies |x_k^k - x^k| \le \|x_k - x\| < \frac 14$$ and since $x_k^k \ge \frac 12$ for all $k$, this implies $x^k > \frac 14$ for all $k \ge N$.

This is incompatible with $x \in \ell_1$, so the answer is NO.