Prove $|\int_\mathbb{R} F(x)\varphi'(x)\,dx|\leq A$ where $F$ has bounded variation on $\mathbb{R}$ and $\varphi\in C^1$.

Question

Let $F$ be of bounded variation on $\mathbb{R}$, i.e it is of bounded variation on any finite subinterval $[a,b]$, $\sup_{[a,b]}V_a^b(F)<\infty$.

Given the fact that \begin{align*} \int_\mathbb{R} |F(x+h)-F(x)|dx \leq A|h|\ \ \text{for all } h\in \mathbb{R} \end{align*} where $A$ is some constant.

Prove that $|\int_\mathbb{R} F(x)\varphi'(x)\,dx|\leq A$ where $F$ has bounded variation on $\mathbb{R}$ and $\varphi\in C^1$ where $\sup_{x\in\mathbb{R}}|\varphi(x)|\leq 1$.

I am not sure how to use the fact provided to deduce the inequality. A hint would be great!