A = 1/5×5 + 1/6×6 + ... + 1/100×100
A < 1/4×5 + 1/5×6 + ... + 1/99×100
A < 1/4 - 1/5 + 1/5 - 1/6 + ... + 1/99 - 1/100
A < 1/4 - 1/100 < 1/4 (1)
A = 1/5×5 + 1/6×6 + ... + 1/100×100
A > 1/5×6 + 1/6×7 + ... + 1/100×101
A > 1/5 - 1/6 + 1/6 - 1/7 + ... + 1/100 - 1/101
A > 1/5 - 1/101 > 1/5 - 1/30
A > 6/30 - 1/30 = 1/6 (2)
Từ (1) và (2) => 1/6 < A < 1/4 ( đpcm)