I'm recently doing some work that's vaguely connected with Dirichlet's approximation theorem. I came across this inequality that I haven't been able to prove.
$$\forall\ a, b \in \mathbb{Z^+}, N\geq1,\ N<b<aN$$
Prove that
$$ \left( \frac{a}{b}\left\lceil \frac{b}{a} \right\rceil \right) < \frac{1}{N}$$
Where $\lceil x \rceil$ denotes the ceiling function of $x$, and $\left( x \right)$ denotes the fractional part of $x$.
I verified the inequality using a computer for small values of $a$, $b$, $N$ and it seems to work, but I can't figure out how to prove it.
(Who uses parentheses to represent fractional part...)
False. Counterexample: \begin{align*} a &= 1117, \\ b &= 133, \text{ and } \\ N &= 294/5 = 58.8 \text{.} \end{align*}
We observe $a,b \in \Bbb{Z}^+$, $N \geq 1$, and $$ N = 58.8 < b = 133 < a N = 65679.6 \text{.} $$
Then \begin{align*} \left\{ \frac{a}{b} \left\lceil \frac{b}{a} \right\rceil \right \} &= \left\{ \frac{1117}{133} \left\lceil \frac{133}{1117} \right\rceil \right \} \\ &= \left\{ \frac{1117}{133} \cdot 1 \right \} \\ &= \left\{ 8.39849{\dots} \right\} \\ &= 0.39849{\dots} \\ &\not\lt \frac{1}{N} \\ &= \frac{5}{294} \\ &= 0.01700{\dots} \text{.} \end{align*}