Find and then prove the supremum of set X

Question

Set X is defined with $$\sqrt {(x (x + 1))}/(2 x + 1)$$, x > 0. I can't prove that for every epsilon greater than zero there exists an t > 0 such that $$1/2 - \sqrt{(t (t + 1))}/(2 t + 1) < \epsilon.$$ Every other step I get.

Answer 1

It is $$\frac{\sqrt{x(x+1)}}{2x+1}\le \frac{1}{2}$$ for $x>0$, this is true, multiplying by $2$ and squaring we get $$4x(x+1)\le 4x^2+4x+1$$ This is true since we get $$0\le 1$$