 #1
mathstudent34
 1
 0
 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
$$\forall x\neq y\in B(0,1),~~~f(x)f(y)(xy)\leq 0.1xy.$$
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
$$f(x)f(y)+xy\leq 0.1xy\implies f(x)f(y)\geq 0.9xy$$
im not sure if this helps.
$$f(x)f(y)+xy\leq 0.1xy\implies f(x)f(y)\geq 0.9xy$$
im not sure if this helps.