Can anyone prove this ?

abdullahg11

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I need to prove that this equation occurs an infinite amount of times for an irrational alpha and Natural n's.
 
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.
 
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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 [nα][n\alpha] stands for the greatest integer function.
It is also known as the floor function and is symbolized as nα\lfloor{n\alpha}\rfloor. SEE HERE
Definition: x\lfloor{x}\rfloor is the greatest integer not exceeding xx.
That means: for all xR\forall x\in\mathbb{R}, for every real number, xZ\bf\lfloor{x}\rfloor\in\mathbb{Z} such that xx<x+1\large\lfloor{x}\rfloor\le x\Large<\lfloor{x}\rfloor +1.
Clearly from the definition: 0xx<10\le x-\lfloor{x}\rfloor<1, for every real number xx.
Now to your specific question: If α\alpha is a irrational real number and nZn\in\mathbb{Z} ten you answer these.
Why is it the case that nαZ ?n\cdot\alpha \notin \mathbb{Z}~?
But why is it the case that nαZ ?\lfloor n\cdot\alpha \in \mathbb{Z}~?
So explain why it the case that 0<nαnα<1 ?0<n\cdot\alpha-\lfloor n\cdot\alpha\rfloor<1~?
You post what you can on the above.
 
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