Inequality trick

[QuantAbu]

Hello everybody, It would be of great help if someone can teach me how they came up with term p/2p [ the one in the red box in the attached pic]. I think its a trick but I don't know what it is. Thank you.

 

[Cubist]

The addends decrease in value

[math] \frac{1}{p+1} > \frac{1}{p+2} > \frac{1}{p+p} [/math]
So if you had "p" of the rightmost term, this would be smaller than the original sum.

[math] \frac{1}{p+1}+\frac{1}{p+2}+\ldots+\frac{1}{p+p} > \frac{1}{p+p} + \frac{1}{p+p} + \ldots+ \frac{1}{p+p}[/math]
Can you simplify the sum on the RHS?

 

[QuantAbu]

Cubist, you have helped me improve my thinking skills. Thank you very much, Of course the RHS simplifies to P/2P. I hope to learn a lot from the people here.