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For each $n=1,2,3,...$ and each $m=0,1,2,...,n-1$, let $$K^n_m = \left[ \frac{3m-n+1}{n} , \frac{3m-n+2}{n} \right] \subset (-1,2). $$ I am struggling these with two questions for quite some time:

(1) $ \ $ The collection $\{ K^n_m \} \setminus \{ K^1_0 \}$ covers the interval $[0,1]$?

In case the answer for (1) is yes, then

(2) $ \ $ For each $ \ x \in [0,1] \ $ is there an infinite number of $ \, K^n_m \, $ such that $ \ x \in K^n_m \ $?

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  • $\begingroup$ Where are you stuck? Do you know what it means for a family of sets $S_j$ to cover $X$? $\endgroup$ – Sean Roberson Aug 14 at 3:44
  • $\begingroup$ @SeanRoberson yes I know the concept. I did not find any kind of regularity that I could use. Just walked in circles. $\endgroup$ – Gustavo Aug 14 at 3:54
  • $\begingroup$ For question (1), do you know what the intervals $K_m^n$ actually are? Did you try working out some examples for small values of $n$? $\endgroup$ – David Aug 14 at 3:58
  • $\begingroup$ @David Sorry! I forgot to exclude the obvious $K^1_0$. Edited now. $\endgroup$ – Gustavo Aug 14 at 4:03

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