Question 101
Let $N$ be a positive integer, $L$ and $x_i$ ($1 \le i \le N$) real numbers such that
$$
0 \le x_1 \lt x_2 \lt x_3 \lt \dots \lt x_N \le L.
$$
What is the least value of $N$ which ensures that at least 10 successive numbers $x_i$ all lie within a distance 1 of each other (i.e. we want $x_i, \dots, x_{i+9}$ so that $x_{i+9} \le x_i + 1$)? Your answer should give $N$ as a function of $L$ and be valid for all $L \gt 0$.
$$
0 \le x_1 \lt x_2 \lt x_3 \lt \dots \lt x_N \le L.
$$
What is the least value of $N$ which ensures that at least 10 successive numbers $x_i$ all lie within a distance 1 of each other (i.e. we want $x_i, \dots, x_{i+9}$ so that $x_{i+9} \le x_i + 1$)? Your answer should give $N$ as a function of $L$ and be valid for all $L \gt 0$.