Ta có:
`1/100 + 1/101 + 1/102 + ... + 1/999 + 1/200`
`= (1/100 + 1/101 + 1/102 + ... + 1/150) + (1/151 + 1/152 + 1/153 + ... + 1/200)`
Lại có:
`1/100 + 1/101 + 1/102 + ... + 1/150 > 1/150 + 1/150 + 1/150 + ... + 1/150`
`=> 1/100 + 1/101 + 1/102 + ... + 1/150 > 1/150 . 50`
`=> 1/100 + 1/101 + 1/102 + ... + 1/150 > 1/3 (1)`
`1/151 + 1/152 + 1/153 + ... + 1/200 > 1/200 + 1/200 + 1/200 + ... + 1/200`
`=> 1/151 + 1/152 + 1/153 + ... + 1/200 > 1/200 . 50`
`=> 1/151 + 1/152 + 1/153 + ... + 1/200 > 1/4 (2)`
Cộng vế với vế lại ta có:
`1/100 + 1/101 + 1/102 + ... + 1/999 + 1/200 > 1/3 + 1/4`
`=> 1/100 + 1/101 + 1/102 + ... + 1/999 + 1/200 > 7/12 (đpcm)`
Vậy `1/100 + 1/101 + 1/102 + ... + 1/999 + 1/200 > 7/12`