Almost Convergent Sequence/Examples/Sequence of alternating zeros and ones converges almost to one half

Theorem
Let $\sequence {x_n}_{n \in \N}$ be the sequence defined by:
 * $x_n = \begin{cases} 0 & : n \equiv 0 \pmod 2 \\ 1 & : n \equiv 1 \pmod 2 \end{cases}$

where $\bmod$ denotes the congruence modulo.

Then $\sequence {x_n}_{n \in \N}$ almost converges to $1/2$.

Proof
Let $\phi$ be a Banach limit.

Let $S$ be the left shift operator on $\map {\ell^\infty} \R$.

Let $\mathbf 1 := \sequence {1, 1, 1, \ldots}$.

Then:

Thus by definition of Banach limit:

That is:
 * $\ds \map \phi {\sequence {x_n} } = \frac 1 2$

As $\phi$ is an arbitrary Banach limit, the claim follows.