1/2^2<1/1*2
1/3^2<1/2*3
1/4^2<1/3*4
....................
1/100^2<1/99*100
⇒1+(1/1*2+1/2*3+1/3*4+...+1/99*100)>A
·1+(1/1*2+1/2*3+1/3*4+...+1/99*100)
=1+(1-1/2+1/2-1/3+1/3-1/4+...+1/99-1/100)
=1+(1-1/100)=1+99/100=1$\frac{99}{100}$
mà 1 99/100>A
do đó A<2 (dpcm)