I am trying to understand the wikipedia article on the Arzelà-Ascoli theorem, which can be found here: https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem
More specifically, I am trying to understand the first of the three examples listed under "Further examples":
To every function $g$ that is $p$-integrable on $[0, 1]$, with $1 < p ≤ \infty$, associate the function $G$ defined on $[0, 1]$ by $$ G(x)=\int_{0}^{x}g(t)\,\mathrm {d} t.$$ Let $F$ be the set of functions $G$ corresponding to functions $g$ in the unit ball of the space $L^p([0, 1])$. If $q$ is the Hölder conjugate of $p$, defined by $1/p + 1/q = 1$, then Hölder's inequality implies that all functions in $F$ satisfy a Hölder condition with $\alpha = 1/q$ and constant $M = 1$. It follows that $F$ is compact in $C([0, 1])$.
To apply Arzelà-Ascoli, we need to show that $F$ is equicontinuous:
A subset $F \subset C(X)$ is said to be equicontinuous if for every $x \in X$ and every $\varepsilon > 0$, $x$ has a neighborhood $U_x$ such that
$$\forall y\in U_{x},\forall f\in {F} : |f(y)-f(x)|<\varepsilon .$$
Now let $G$ be a function in $F$ corresponding to $g$. Then we want to bound $| G(y) - G(x) |$ independent from $G$ itself. If I wanted to apply Hölder's inequality, I'd go for the following (wlog. $x\leq y$): $$ |G(y)-G(x)| = \left|\int_0^1 g(\xi)\cdot\mathbb{1}_{[x,y]}(\xi) \,\mathrm d\xi \right| \leq \left\lVert g \right\rVert_p \cdot \left\lVert \mathbb{1}_{[x,y]} \right\rVert_q = \left\lVert g \right\rVert_p (y-x)^{1/q}, $$ which is a Hölder condition with $\alpha=1/q$ and "constant" $M=\left\lVert g \right\rVert_p$, not $1$. Am I doing something wrong? How can I get rid of this $\left\lVert g \right\rVert_p$ factor, which could potentially blow up arbitrarily?