On a Riemannian manifold $(M, g)$ , we have a function $\psi : M \to \Bbb R$, which is differentiable, and furthermore satisfies some "small derivative" condition as follows : if we consider the one-form given by $d\psi$, then since $g$ is a metric, the quantity $g(d\psi,d\psi) := (||d\psi||_g)^2$ makes sense and is an element of $\mathbb R$. I am asked to assume that $||d\psi||_g \leq \frac 14$.
From here I would like to find a smooth function $\psi_0 : M \to \Bbb R$ such that $|\psi - \psi_0| \leq 1$ and $||d\psi_0||_g \leq 1$.
My first approach to this problem was to seek a solution for Euclidean space with the Euclidean inner product. Here, I see only convolution with a suitable mollifier at some point as a solution. But can this even be carried over to a manifold?
Furthermore, I do not quite know how a small change in the function changes $d\psi$ and therefore $||d\psi||_g$, so my ability to attack this problem is just about as blunt as it gets.
EDIT 1 : For further context, this is part of an exercise which is used to prove an extension of the Hopf-Rinow theorem (there is a version which includes two more conditions than the usual, and is apparently useful in the field of several complex variables to prove one of their "big" theorems, the name of which I don't happen to know unfortunately).
EDIT 2 : Nash's embedding theorem tells us that without loss of generality we can assume that $(M,g)$ is just a submanifold of some Euclidean space with the restriction of the Euclidean metric, but I don't want to use this.