prove that 1/2 * 1/3 * 1/4 * ... ..... ... ..... ..... .... ....... .... ... ....1/99 < 1/2

Answer 1

1/2 * 1/3 < 1/2

all the rest of the fractions are less than 1, so the more of them you multiply by, the smaller the product gets.

you have 1/2 * a*b*c*d*... where each of a,b,c,d,... < 1

or, you have 1/99! = 1/2 * 2/99! < 1/2