Limits if polar coordinates (conceptual explanation)

[negation]

1. Homework Statement [/b]

Suppose the lim(x,y) →(0,0) (xy)/SQRT[x^2 + y^2] if it exists

find the limit.

The Attempt at a Solution

x = rcosΘ
y = r sinΘ
r = SQRT[x^2 + y^2]

∴ lim_{r → 0} (r²cosΘrsinΘ)/ r = rcosΘsinΘ \leq r

and so -r \leq(xy)/SQRT[x^2 + y^2] \leq r

...
...

I can understand the limit goes to zero because algerbraic multiplication and the sandwhich theorem tells me so. However, the highlighted part in red confuses me. What is the significance of
rcosΘsinΘ \leqr?

 

[UltrafastPED]

It was to generate the inequality - they used cosΘrsinΘ ≤ 1.