Construct a sequence $(x_{n})$ such that $\bigcap_{n=1}^{\infty} K_{n} = \{x\}$, and $(x_{n})$ is not bounded.

Question

Let $(x_{n})$ be a sequence in $\ell^{p}$ for some $1 < p < \infty$ and let $x \in \ell^{p}$ be given. Set $$ K_{n} = \overline{\text{conv}\left( \bigcup_{i=n}^{\infty} \{x_{i}\} \right)}. $$

Construct a sequence $(x_{n})$ such that $\bigcap_{n=1}^{\infty} K_{n} = \{x\}$, and $(x_{n})$ is not bounded.

Answer 1

Let $e_k$ be the sequence that is $1$ in its $k$-th term and $0$ everywhere else.

Define $x_{2k-1}=x+k\cdot e_k$ and $x_{2k}=x-k\cdot e_k$. Then $x\in K_n$ for all $n$, and $(x_n)$ is not bounded. I think this works?