Power Series: the Sequence of Terms, the Coefficients 1/(k-1)!, and the Sum as a Map on the Set of Convergence

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(c_{k})$ be a sequence in $\mathbb{R}$. For $u\in\mathbb{R}$ and $n\in\mathbb{N}_{0}$, $u^{n}$ is the power of Powers with Exponents in the Natural Numbers with Zero (i) and Powers with Exponents in the Natural Numbers with Zero (ii), so that $u^{0}=1$; for $k\in\mathbb{N}$, $k-1\in\mathbb{N}_{0}$ is the difference; the factorial $n!$ of $n\in\mathbb{N}_{0}$, read in $\mathbb{R}$, is its image as in Commutative Rings, Fields and Ordered Fields: Standard Notation (iv), and is nonzero by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii); and convergence of a series and its sum are as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii).

(i) For every $u\in\mathbb{R}$ there is exactly one sequence in $\mathbb{R}$ whose $k$-th term is $c_{k}u^{k-1}$ for every $k\in\mathbb{N}$.

(ii) There is exactly one sequence in $\mathbb{R}$ whose $k$-th term is $1/(k-1)!$ for every $k\in\mathbb{N}$.

(iii) Let $E\subseteq\mathbb{R}$ be such that, for every $u\in E$, the series $\sum_{k=1}^{\infty}c_{k}u^{k-1}$ of the sequence of (i) converges. Then there is exactly one map $h:E\to\mathbb{R}$ with $h(u)=\sum_{k=1}^{\infty}c_{k}u^{k-1}$ for every $u\in E$.