= Solution
For $y=x-Sx\in Y$, the telescoping arithmetic means satisfy
$$
\frac1N\sum_{n=1}^Ny_n=\frac{x_1-x_{N+1}}N\longrightarrow0,
$$
because $x$ is a <bounded sequence>. Hence $\|\mathbf1-y\|_\infty\geq|1-N^{-1}\sum_{n=1}^Ny_n|$ for every $N$, and passage to the limit gives $\|\mathbf1-y\|_\infty\geq1$. Taking $y=0$ shows $d(\mathbf1,Y)=1$. The proved <Hahn-Banach distance formula> therefore yields a <bounded linear functional> $L$ on the real <l-infinity sequence space> with
$$
\boxed{\|L\|=1,\qquad L(\mathbf1)=1,\qquad L|_Y=0.}
$$
We verify that this single <linear functional> also has the two remaining properties.
Back to article page
