Why is the following statement correct?(It's part of the proof of the inverse function theorem)
Let $f:\underbrace{U}_{open} \subset \mathbb{R}^n \to \mathbb{R}^n$ be continuously differentiable and $f'(x_0)$ invertible in $x_0 \in U$.
Then there exists $R>0$, such that for all $x \in U$ with $||x-x_0||\leq R$ there holds: $$ ||I_n-f'(x_0)^{-1}*f'(x)||\leq \theta <1. $$
($I_n$ is the unit Matrix)
Suppose the result is not true for any $R$ (with $\theta =\frac 1 2$). Then we can find $(x_j)$ such that $\|x_j-x_0\| <\frac 1 n$ and $\|I_n- f'(x_0)^{-1} f'(x_j)\| \geq \frac 1 2$. We have $f'(x_j) \to f'(x_0)$ and hence $f'(x_0)^{-1} f'(x_j) \to f'(x_0)^{-1}(f'(x_0))=I_n$. But then $\|I_n- f'(x_0)^{-1} f'(x_j)\| \to 0$. This is a contradiction. [All convergences are in the norm.].