Theorem 8.8.2 (Successive Approximations [Section III.2, SW99]). Let $E, F$ be metric TVSs over $K \in \RC$ with pseudonorm $\rho$ and $\eta$, respectively. Let $T \in L(E; F)$, $r > 0$, $\gamma \in (0, 1)$, and $C \ge 0$. Suppose that for every $y \in B_{F}(0, r)$, there exists $x \in E$ such that:
$\eta(y - Tx) \le \gamma \eta(y)$.
$\rho(x) \le C \eta(y)$.
then for any $y \in F$, there exists $\seq{x_n}\subset E$ such that:
$\sum_{n \in \natp}\rho(x_{n}) \le C\eta(y)/(1 - \gamma)$.
$y = \limv{N}\sum_{n = 1}^{N} Tx_{n}$.
In particular,
Proof. Let $y_{0} = y$ and $x_{0} = 0$. Let $N \in \natz$ and suppose inductively that $\seqf[N]{x_n}\subset E$ has been constructed such that:
$\sum_{n = 1}^{N}\rho(x_{n}) \le C\eta(y)\sum_{n = 0}^{N-1}\gamma^{n}$.
$\eta\paren{y - \sum_{n = 1}^N Tx_n}\le \eta(y)\gamma^{N}$.
By assumption, there exists $x_{N+1}\in E$ such that:
$\eta\paren{y - \sum_{n = 1}^{N+1} Tx_n}\le \gamma \eta\paren{y - \sum_{n = 1}^N Tx_n}\le \gamma^{N+1}$.
$\rho(x_{N+1}) \le C\eta\paren{y - \sum_{n = 1}^N Tx_n}\le C\eta(y)\gamma^{N}$.
Combining (I) and (ii) shows that $\sum_{n = 1}^{N} \rho(x_{n}) \le C \eta(y) \sum_{n = 0}^{N} \gamma^{n}$. Therefore there exists $\seq{x_n}\subset E$ such that (I) and (II) holds for all $N \in \natp$.
By (I), $\sum_{n \in \natp}\rho(x_{n}) \le C\eta(y)\sum_{n \in \natz}\gamma^{n} = C \eta(y)/(1 - \gamma)$. By (II), $\limv{N}\eta\paren{y - \limv{N}\sum_{n = 1}^N Tx_n}= \limv{N}\eta(y)\gamma^{N} = 0$.$\square$