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$g(x)\xrightarrow{x\to\infty}\infty$ Implies $g'(x)\leq g^{1+\varepsilon}(x)$

Recently in my ordinary differential equations class we were given the following problem:

>Suppose $g:(0,\infty)\to\mathbb{R}$ is an increasing function of class $C^{1}$ such that
$g(x)\xrightarrow{x\to\infty}\infty$. Show that for every $\varepsilon>0$ the inequality $g^{\prime}(x)\leq g^{1+\varepsilon}(x)$ upholds,
outside a set of finite length.

I thought about using [Grönwall's inequality][1] , but i did not got any useful result.

[1]: https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality