√n - √(n - 1) < 0.01
√n < 0.01 + √(n - 1)
Take squares on both sides and simplify:
n < 0.0001 + 0.02 √(n - 1) + (n - 1)
0.9999 < 0.02 √(n - 1)
49.995 < √(n - 1)
Take squares on both sides again:
2499.5 < n - 1
2500.5 < n
So we must have n ≥ 2501 for the inequality to hold.
