Prove this equation involving Landau Notation

Question

I started working through this work about number theory, but I've never really used Landau notation before so I am sometimes struggling to understand the steps used in the proofs. Specifically in Theorem $1.8$ in the last equation I believe it is asserted that $$ \int_Y^\infty\frac{u-\lfloor u \rfloor}{u^2}du = \mathcal{O}(\frac{1}{Y}) $$ How can I see this to be the case?

Answer 1

It's between $0$ and $\int_Y^\infty\tfrac{1}{u^2}du=\tfrac1Y$.