Let $C_0(\mathbb{R})$ be the space of $\mathbb{R}$ valued sequences converging to $0$. Let $l_n$ be a positive sequence in $\mathbb{R}$ such that $\sum\limits_{n=1}^\infty l_n=1$. We define $ F:C_0(\mathbb{R})\to\mathbb{R}:x\mapsto\sum\limits_{n=1}^\infty x_n l_n. $ Further, define $M = \{x \in C_0(\mathbb{R}) : F(x) = 0\}$.
We are supposed to prove:
1) for $x \in C_0(\mathbb{R})$ and $y \in M\setminus\{x\}$, we have $|F(x)|\leq \sup\limits_{n\in\mathbb{N}}|x_n-y_n|$.
2) for $x \in C_0(\mathbb{R})$ we have $\mathrm{dist}(x,M)\leq|F(x)|$.
Our teacher gave us a hint for the second part which is to consider $x - F(x)z$, where $z$ is the constant $1$ sequence. This is not in $M$ but can be approximated in a suitable way.
How does one tackle this problem?
Thank you.