# Proving that a function is Riemann-stieltjes integrable

Let $$g$$ a increasing function, and $$f$$ integrable with respect to $$g$$ in $$J=[a,b]$$ proof that $$|f|$$ is integrable with respect to $$g$$

By definition if $$f$$ is integrable with respect to $$g$$, for every partition $$P$$ of $$J$$ and for every $$\epsilon>0$$ exists $$I$$ in $$R$$ such that if $$Q$$ refine $$P$$ then $$|S(Q,f,g)-I|< \epsilon$$ $$|S(Q,f,g)-I|= \sum^{n}_{j=0}f(\lambda_{j})(g(x_{j}-x_{j-1})$$ then

$$\sum^{n}_{j=0}|f(\lambda_{j})|(g(x_{j}-x_{j-1}) \geq \sum^{n}_{j=0}f(\lambda_{j})(g(x_{j}-x_{j-1})$$

but i don’t know how to do

$$\epsilon \geq \sum^{n}_{j=0}|f(\lambda_{j})|(g(x_{j}-x_{j-1})$$