Can anyone prove this ?

[abdullahg11]

I need to prove that this equation occurs an infinite amount of times for an irrational alpha and Natural n's.

 

[Deleted member 4993]

View attachment 18693
I need to prove that this equation occurs an infinite amount of times for an irrational alpha and Natural n's.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[abdullahg11]

I need to prove that there are infinite natural numbers that solve the equation. I've ran a code and it seems to be right.

 

[Probability]

Please post your results..

 

[pka]

View attachment 18693
I need to prove that this equation occurs an infinite amount of times for an irrational alpha and Natural n's.

You should not only post your attempts at a solution but tell us the definitions of symbols.
For example: I assume that \([n\alpha]\) stands for the greatest integer function.
It is also known as the floor function and is symbolized as \(\lfloor{n\alpha}\rfloor\). SEE HERE
Definition: \(\lfloor{x}\rfloor\) is the greatest integer not exceeding \(x\).
That means: for all \(\forall x\in\mathbb{R}\), for every real number, \(\bf\lfloor{x}\rfloor\in\mathbb{Z}\) such that \(\large\lfloor{x}\rfloor\le x\Large<\lfloor{x}\rfloor +1\).
Clearly from the definition: \(0\le x-\lfloor{x}\rfloor<1\), for every real number \(x\).
Now to your specific question: If \(\alpha\) is a irrational real number and \(n\in\mathbb{Z}\) ten you answer these.
Why is it the case that \(n\cdot\alpha \notin \mathbb{Z}~?\)
But why is it the case that \(\lfloor n\cdot\alpha \in \mathbb{Z}~?\)
So explain why it the case that \(0<n\cdot\alpha-\lfloor n\cdot\alpha\rfloor<1~?\)
You post what you can on the above.