Prove $\frac{1}{2}+\frac{1}{2(u+1{)}^{2}}-\frac{1}{\sqrt{1+2u}}\ge 0$ for $u\ge 0$

This inequality provides a tight lower bound to $\sqrt{1+2u}$ for $u\ge 0$ without a radical. I was trying to solve it by squaring the radical and cross-multiplying and repeated differentiation of the resulting expression, I wonder if there is a quicker solution. Thanks.

This inequality provides a tight lower bound to $\sqrt{1+2u}$ for $u\ge 0$ without a radical. I was trying to solve it by squaring the radical and cross-multiplying and repeated differentiation of the resulting expression, I wonder if there is a quicker solution. Thanks.