I need help to prove the following principle-
Any identity between real or complex power series, involving addition, multiplication (possibly infinite sums and products), and substitution, is an identity in the ring of formal power series.
Answer:
Suppose $f(x)=\sum_{n \geq 0} a_nx^n$ and $g(x)=\sum_{n \geq 0} b_nx^n$ be two power series in $ \mathbb{R}$.
Also let the identity $ f(x)=g(x)$ holds in $\mathbb{R}$.
Then we have to show that same identity $f(X)=g(X)$ i.e., $ \sum_{n \geq 0} a_nX^n=\sum_{n \geq 0} b_nX^n$ holds in the ring of formal power series $\mathbb{R}[[X]]$.
Let $ h(x)=f(x)-g(x)=\sum_{n \geq 0} a_nx^n-\sum_{n \geq 0} b_nx^n=0$.
We know that Taylor series of an analytic function is unique.
Does this conclude the proof?
Help me to prove the principle with any further requirements.
Hint: $a_n = \dfrac{f^{(n)}(0)}{n!}$ holds in the ring of analytic functions on $\mathbb R$ and in the ring of formal power series $\mathbb{R}[[X]]$.