Problem $2$

We are given $n$ real numbers in increasing order: $x_{i}1inx=[0\mathrm{.}3,0\mathrm{.}5,0\mathrm{.}6,1\mathrm{.}4,1\mathrm{.}6,2\mathrm{.}3,3\mathrm{.}8]\{[0\mathrm{.}2,1\mathrm{.}2],[1\mathrm{.}3,2\mathrm{.}3],[3\mathrm{.}0,4\mathrm{.}0]\}{x}_1.We$ would like $to$ choose the minimum number $of$ unit intervals $(intervalsof$ length one$)$ that cover all $of$ the input numbers, $i.e.,we$ would like $x_{i}tobe$ contained $inat$ least one $of$ the chosen intervals for all $1in.$

For example, $ifwe$ are given $x=[0\mathrm{.}3,0\mathrm{.}5,0\mathrm{.}6,1\mathrm{.}4,1\mathrm{.}6,2\mathrm{.}3,3\mathrm{.}8]$ then $\{[0\mathrm{.}2,1\mathrm{.}2],[1\mathrm{.}3,2\mathrm{.}3],[3\mathrm{.}0,4\mathrm{.}0]\}isan$ optimal solution with $3$ intervals.

State and prove the greedy choice property for this problem.