An Application: Banach Limits

If $L$ is a Banach limit, show that there are $x$ and $y$ in $l^\infty$ such that $L(xy) \neq L(x)L(y)$.

Let $X$ be a set and $\Omega$ a $\sigma$-algebra of subsets of $X$. Suppose $\mu$ is a complex-valued countably additive measure defined on $\Omega$ such that $\left\| \mu \right\| = \mu (X) < \infty$. Show that $\mu (\Delta ) \geq 0$ for every $\Delta$ in $\Omega$. (Though it is difficult to see at this moment, this fact is related to the proof of (c) in Theorem 7.1 for the complex case.)

Show that if $x \in l^\infty$, $\left\| x\right\|_{\infty } \leq 1$, then there is a sequence $\left\{ x_n\right\}$, $x_n$ in $l^\infty$ such that $\left\| x_n\right\|_{\infty } \leq 1$, $\left\| x_n - x\right\|_{\infty } \to 0$, and each $x_n$ takes on only a finite number of values.