Fréchet Space (Functional Analysis) is Complete Metric Space

Theorem
Let $\struct {\R^\omega, d}$ be the Fréchet space on $\R^\omega$.

Then $\struct {\R^\omega, d}$ is a complete metric space.

Proof
From Fréchet Space (Functional Analysis) is Metric Space, $\struct {\R^\omega, d}$ is a metric space.

It remains to be demonstrated that $\struct {\R^\omega, d}$ is complete.

Let $\sequence {x^n}_{n \mathop \in \N}$ be an arbitrary Cauchy sequence in $\R^\omega$, where as an exception we denote the index of the Cauchy sequence as a superscript.

Recall that each $\forall n \in \N: x^n \in \R^\omega$ itself is a sequence in $\R$ by the definition of $\R^\omega$.

More precisely, each $x^n \in \R^\omega$ can be written as
 * $x^n := \sequence { {x^n}_i }_{i \mathop \in \N}$

where ${x^n}_i \in \R$ for all $i \in \N$.

Lemma
For each $j \in \N$, the real sequence $ \sequence { {x^n}_j}_{n \mathop \in \N} $ is convergent.

Proof
Let $j \in \N$.

In view of Cauchy's convergence criterion, it suffices to show that $\sequence { {x^n}_j} _{n \mathop \in \N} $ is a Cauchy sequence in $\R$.

Let $\epsilon > 0$ be arbitrary.

Since $ \sequence {x^n} _{n\in\N} $ is a Cauchy sequence in $\R^\omega$, there is a $N \in \N$ such that:
 * $\forall m, n \in \N: m, n \ge N: \map d {x^m, x^n} < \dfrac {2^{-j} \epsilon} {1 + \epsilon}$

Thus:
 * $\dfrac {2^{-j} \size {{x^m}_j - {x^n}_j} } {1 + \size {{x^m}_j - {x^n}_j} } \le \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {{x^m}_i - {x^n}_i} } {1 + \size {{x^m}_i - {x^n}_i} } = \map d {x_m, x_n} < \dfrac {2^{-j} \epsilon} {1 + \epsilon}$

which implies:
 * $\size { {x^m}_j - {x^n}_j} < \epsilon$

Altogether:
 * $\forall m, n \in \N: m, n \ge N: \size {{x^m}_j - {x^n}_j} < \epsilon$

Let $\ds x^\infty := \sequence {{x^\infty}_i}_{i \mathop \in \N} \in \R^\omega$, where $\ds {x^\infty}_i := \lim_{n \mathop \to \infty} {x^n}_i$ exists for each $i \in \N$ by the preceding lemma.

It remains to show that $x^\infty$ is the limit of $\sequence {x^n}_{n \in \N}$ in $\R^\omega$.

To this end, let $\epsilon > 0$ be arbitrary.

For each $i \in \N$, there is a $N_i \in \N$ such that:
 * $\forall n \ge N_i: \size { {x^\infty}_i - {x^n} _i} < \dfrac \epsilon 4$

Furthermore, $M \in \N$ be so large that $\dfrac 1 {2^M} < \dfrac \epsilon 2$.

Finally, let $N := \max \set {N_0, N_1, \ldots, N_M}$.

Then, for all $n \ge N$: