And also, what are some techniques to show such a thing. That is, what are standard techniques for showing a fraction is less than one? I'm just trying to refresh/improve/add to my toolbox.
The fraction I am working with is: $$ \frac{tx(tx+1) +t^2}{(tx+1)^2} $$ The problem is: Show $$ \frac{tx(tx+1) +t^2}{(tx+1)^2} \leq 1 \text{ for } x \geq t,\ t\geq 0 $$
I have been trying to move the denominator to the right and show that the numerator is less than the denominator, but I am not seeing any clean cancellations, or obvious result.
Since $x\ge t\ge0$, we have $tx\ne-1$ and so $$\eqalign{t^2\le tx\quad &\Rightarrow\quad t^2\le tx+1\cr &\Rightarrow\quad tx(tx+1)+t^2\le tx(tx+1)+tx+1\cr &\Rightarrow\quad tx(tx+1)+t^2\le (tx+1)^2\ .\cr}$$