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I'm trying to understand the proof of the inverse function theorem in Rudin's Principles of Mathematical Analysis:

Let $E\subset{\Bbb R}^n$ be an open set, and $f:E\to{\Bbb R}^n$ be a $C^1$-mapping. Suppose $f'(a)$ is invertible for some $a\in E$ and $b=f(a)$. Then there exist open set $U$ and $V$ in ${\Bbb R}^n$ such that $a\in U$ and $b\in V$, $f$ is a bijection from $U$ to $V$. Also $f^{-1}\in C^1(V)$.

As I see from the book, a key point of the proof is the construction of the following contraction mapping: $$ \varphi_y(x)=x+[f'(a)]^{-1}(y-f(x)),\quad x\in E $$

Here are my questions:
What's the motivation of such construction? How can one come up with such construction naturally?

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    I've just found that this is closely related to http://math.stackexchange.com/q/433283/9464 –  Aug 25 '13 at 14:37
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    $\psi_y(x) = x + [f'(x)]^{-1}(y - f(x))$ is the Newton iteration to find the zero of $y - f(x)$. Fixing the point of the derivative as $a$ makes the convergence slower, but makes the entire thing easier to analyse. – Daniel Fischer Aug 25 '13 at 14:38

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