Consider the series , where a is a positive real number.
$a, 2a, 3a, .... (n-1)a$
Prove that there is one member of this series that differs from an integer by at most $\frac{1}{n}$
My approach :
Draw a modular circle of $1$, around which the number line goes infinitely often. The radius of this circle is $\frac{1}{2\pi}$. All the points are now plotted onto this circle.
Let $C$ be the point on the circle which represents $ 0$,and then all the integers(which give a remainder of $0 $with division by $1$).The length of the arc $ CM $is defined as
$CM - [CM] $where $[CM] $is the floor function
I managed to reduce this to a simpler problem.
To prove that any two points,
${ja - ia \leq \frac{1}{n}}$
${(j-i)a \leq \frac{1}{n}}$
$ka \leq 1/n$
So I divided the circle into $\frac{1}{n}$ parts and thought of using the Pigeonhole Principle to solve it but there are only $(n-1)$ members of the series. I can't apply it.
How can I solve this ?