ILL 1988 Problem 2

Let $\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots,$ where $[x]$ denotes the integer part of $x.$ Prove that

i.) there are infinitely many positive integers $m$ such that $a_{m+1} - a_m > 1;$ ii.) there are infinitely many positive integers $m$ such that $a_{m+1} - a_m = 1.$