hi all, i've got a question as well as the solution but can anyone kindly explain the solution or kindly set up an example for me? thank you all very much in advance.
Let there be 5 distinct real numbers a1-a5. Prove there are indices i,j with 0<ai-aj<1+aiaj
As the function tan-pi/2,+pi/2) ->R is surjective, there are angles θi belongs to (-pi/2,+pi/2) with ai = tan θ, 1<=i<=5.
Divide the interval (-pi/2,+pi/2) into 4 equal pieces, each of length pi/4. As we have 5 angles, at least 2 of them must lie in the same small interval, implying that there are i,j with 0<θi-θj<pi/4. Applying tan to the last inequality and using the identity
tan(x-y)=(tan x - tan y) / 1 + tan x tan y