how you doing guys? I'm a new member of this wonderful forum. I can't thank you enough. I'm Italian, so forgive me for some english mistakes I may do. I'm translating a problem, so I hope to do it in the best way. Here we are...
Let $\tau$ = $\frac{1+\sqrt5}{2}$ Note that $\tau + 1 = \tau^2$
Let $a_n$=$\lfloor n\tau \rfloor$, $b_n$=$\lfloor n\tau^2\rfloor$ for every natural number
a) Prove that for every number $N>0$ there are $\lfloor\frac{(N+1)}{\tau}\rfloor a_n$ elements and $\lfloor\frac{(N+1)}{\tau^2}\rfloor b_n$ elements which are $\le N$.
b) Prove that: $N-1<\lfloor{\frac{(N+1)}{\tau}}\rfloor + \lfloor{\frac{(N+1)}{\tau^2}}\rfloor < N+1$
c) Demonstrate that every natural number belongs to $a_n$ or to $b_n$, and that $a_n$ and $b_n$ don't have any common elements.
I have few ideas on how to demonstrate (a) and (b)... I would be very grateful if someone could help me. Thanks a lot in advance.