Ta có : `1/51+1/52+1/53+......+1/100`

`=(1/51+1/52+.....+1/60)+(1/61+1/62+......+1/70)+(1/71+1/72+......+1/80)+(1/81+1/82+......+1/90)+(1/91+1/92+......+1/100)`

`->S>(1/60+1/60+.......+1/60)+(1/70+1/70+.....+1/70)+(1/80+1/80+.....+1/80)+(1/90+1/90+.....+1/90)+(1/100+1/100+.....+1/100)`

`->S>1/60*10+1/70*10+1/80*10+1/90*10+1/100*10`

`->S>1/6+1/7+1/8+1/9+1/10`

`->S>1627/2520`

Mà `1627/2520>7/12` nên `S>7/12 (1)`

Ta có : 

`1/51+1/52+1/53+......+1/100`

`=(1/51+1/52+.....+1/59)+(1/60+1/61+......+1/69)+(1/70+1/71+......+1/79)+(1/80+1/81+......+1/89)+(1/90+1/91+......+1/99)`

`->S<(1/51+1/51+.....+1/51)+(1/60+1/60+.....+1/60)+(1/70+1/70+.....+1/70)+(1/80+1/80+.....+1/80)+(1/90+1/90+.....+1/90)`

`->S<1/51*9+1/60*10+1/7*10+1/8*10+1/9*10`

`->S<9/51+1/6+1/7+1/8+1/9`

`->S<6187/8586`

Mà `6187/8586<5/6` nên `S<5/6` `(2)`

Từ `(1)(2)->đpcm`