I'm studying some basic notions of Young Integration and I got stuck with the so called "Link between $p-variation$ and $Holder$ norm" found on the wikipedia page: the criminal page.
In particular it says that if $f$ has finite $p-variation$ and it is continuous than there exists a reparameterization $\tau$ such that $f \circ \tau$ is $\frac{1}{p}-Holder$.
Any suggestions to show the existence of such a reparameterization?
Considering just functions on $[0,\infty)$ for convenience: Let $v(x)$ be the $p$-variation of $f$ on $[0,x]$. Then for $0<x<y$, $v(y)-v(x)$ is the $p$-variation of $f$ on $[x,y]$, so that $$|f(y)-f(x)|^p\le v(y)-v(x).$$
Now let $u(x)=x+v(x)$; note that if $x<y$ then $v(y)-v(x)\le u(y)-u(x)$. Since $u$ is continuous and strictly increasing it has a continuous inverse $\tau$; now$$|f(\tau(y))-f(\tau(x))| \le|v(\tau(y))-v(\tau(x))|^{1/p}\le|u(\tau(x))-u(\tau(y))|^{1/p}=|y-x|^{1/p}.$$
