show that $m^*(f(E))\le Km^*(E)$

Question

If $f:\Bbb R\to \Bbb R$ and satisfies $|f(x)-f(y)|\le K|x-y|$ forall $x$ and $y$,show that $m^*(f(E))\le Km^*(E)$.

Let $\epsilon>0$ be given ,then $\exists \{I_n\}$ such that $E\subset \cup I_n$ and $m^*(E)+\epsilon>\sum l(I_n)$.

If I can show that $f(E)\subset K(\cup I_n)$ then

Since $f(E)\subset K(\cup I_n)\implies m^*(f(E))\le K\sum l(I_n)

But I am unable to show that $f(E)\subset K(\cup I_n)$.Please help me to show that .

Answer 1

You cannot really show that, but rather $f(E) \subset \cup f(I_n)$ and then notice that $l(f(I_n)) \le K l(I_n)$ and you are almost done ( the constant obtained on RHS is $K \cdot \epsilon$).