Ta có:
99 - (1/2 + 1/3 + 1/4 + ... + 1/100)
= 99 - (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100 + 1/100)
= 99 - 1
= 98

Và:
1/2 + 2/3 + 3/4 + ... + 99/100
= (1 - 1/2) + (2/3 - 1/2) + (3/4 - 2/3) + ... + (99/100 - 98/99)
= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100
= 1 - 1/100
= 99/100

Vậy chúng ta đã chứng minh được rằng:
99 - (1/2 + 1/3 + 1/4 + ... + 1/100) = 1/2 + 2/3 + 3/4 + ... + 99/100.