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I'm reading Wendell Fleming's book Functions of Several Variables, and on page 144, he states the following lemma:

Suppose $\phi: \mathbb{R}^n \to \mathbb{R}^n$ is continuous on some neighborhood $\Omega$ of $0$, and such that

$$\|\phi (t)\| \leq c \|t\| \; \; \; \forall t \in \Omega, \text{ where }0<c<1.$$

Define $\Psi(t) := \sum_{i=0}^\infty \phi^{[i]}(t)$ and $\Psi_r(t) := \sum_{i=0}^r \phi^{[i]}(t)$, where $\phi^{[i]}(t):= \phi \circ \dotsb \circ \phi$ is the i-fold composition of $\phi$. Then

  1. $\|\Psi(t)\| \leq \frac{|t|}{1-c}$
  2. $\Psi(t) - \phi[\Psi(t)]=t$.

In proving the second part, he states:

$$\Psi_r - \phi \circ \Psi_r = \sum_{i=0}^r \phi^{[i]} - \sum_{i=1}^{r+1} \phi^{[i]}= I - \phi^{[r+1]}.$$

I cannot figure out why $\Psi_r - \phi \circ \Psi_r = \sum_{i=0}^r \phi^{[i]} - \sum_{i=1}^{r+1} \phi^{[i]}$ holds. It seems it would be true for $\Psi_r - \Psi_r \circ \phi$. I'm sure I'm just missing something obvious, but I can't see why it is true.

I checked to make sure Fleming's notation of function composition is the usual one, and it is. I checked to make sure it wasn't a typo...the rest of the proof proceeds the same way. What am I missing?

Here is a scan of the page, if it helps: https://i.stack.imgur.com/Ufib4.jpg

Eric Auld
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1 Answers1

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First note that $\| \Phi^{[i]}(t)\|\leq c^i\|t\|$ (it follows from an easy induction). Then the inequality follows from $0<c<1$ and that $\sum_{i=1}^\infty c^i\|t\|$ is a geometric progression.

azarel
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