Convergent Sequences form Invariant Subspace of Bounded Sequences wrt Cesàro Summation Operator

Theorem
Let $\ell^\infty$ be the space of bounded sequences.

Let $c$ be the space of convergent sequences.

Let $A : \ell^\infty \to \ell^\infty$ be the Cesàro summation operator.

Then $c$ is an invariant subspace of $\ell^\infty$ $A$.

Proof
Let $\sequence {x_n}_{n \mathop \in \N} \in c$ be a sequence.

By definition, $\sequence {x_n}_{n \mathop \in \N}$ converges.

Let $\ds L = \lim_{n \mathop \to \infty} x_n$ be the limit of $\sequence {x_n}_{n \mathop \in \N}$.

Then:


 * $\forall \epsilon \in \R_{> 0} : \exists N_1 \in \N : \forall n \in \N : n > N_1 \implies \size {x_n - L} < \epsilon$

Furthermore:


 * $\exists M \in \R_{> 0} : \forall n \in \N : \size {x_n} < M$

$A \mathbf x$ converges to $L$
Let $\epsilon' \in \R_{> 0 }$.

Let $n \in \N$ be such that:


 * $\ds \frac {N_1} n \paren{M + \size L} + \paren {1 - \frac {N_1} n} \epsilon < \epsilon'$

Suppose $n > N_1$.

Then:

Suppose also that $\ds n > N_1 \frac {M + \size L}{\epsilon' - \epsilon}$.

Then:

Altogether, $n$ has to satisfy $n > \tilde N$ where:


 * $\ds \tilde N = \max \set {N_1, N_1 \frac {M + \size L}{\epsilon' - \epsilon} }$.

$\epsilon'$ was arbitrary.

Hence:


 * $\ds \forall \epsilon' \in \R{>0} : \exists \tilde N : \forall n \in \N : n > \tilde N \implies \size {A \mathbf x - L} < \epsilon'$

Therefore, $A \mathbf x \in c$.

Thus, $A c \subseteq c$.

By definition, $c$ is an invariant subspace of $\ell^\infty$ $A$.