- #1

mathstudent34

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- Homework Statement
- Let $B(0,1)\subseteq\mathbb{R^n}$ be the ball of radius $1$ in $\mathbb{R^n}$. Suppose $f:B(0,1)\to\mathbb{R^n}$ satisfies $f(0)=0$ and

$$\forall x\neq y\in B(0,1),~~~|f(x)-f(y)-(x-y)|\leq 0.1|x-y|.$$

Show that $f$ is onto $B(0,0.4)$.

- Relevant Equations
- $f:X\to Y$ is surjective if $\forall y\in Y,\ \exists x\in X$ such that $f(x)=y$.

I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality

$$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$

im not sure if this helps.

$$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$

im not sure if this helps.