# Collection of intervals covers $[0,1]$?

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 \$$?

• Where are you stuck? Do you know what it means for a family of sets $S_j$ to cover $X$? – Sean Roberson Aug 14 at 3:44
• @SeanRoberson yes I know the concept. I did not find any kind of regularity that I could use. Just walked in circles. – Gustavo Aug 14 at 3:54
• 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 $n$? – David Aug 14 at 3:58
• @David Sorry! I forgot to exclude the obvious $K^1_0$. Edited now. – Gustavo Aug 14 at 4:03