Nested fractions

We can easily verify that $\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}}} \approx 1.04$ so $\forall x > 0, \sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+x}} > 1$

$\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\sqrt{\frac{1}{4}+\sqrt{...}}}}<1$ $\Leftrightarrow \frac{1}{2}+\sqrt{\frac{1}{3}+\sqrt{\frac{1}{4}+\sqrt{...}}}<1^2$ $\Leftrightarrow \sqrt{\frac{1}{3}+\sqrt{\frac{1}{4}+\sqrt{...}}}<\frac{1}{2}$ $\Leftrightarrow \frac{1}{3}+\sqrt{\frac{1}{4}+\sqrt{...}}<\frac{1}{4}$ $\Leftrightarrow \sqrt{\frac{1}{4}+\sqrt{...}} <-\frac{1}{12}$ The left-hand-side is positive, which can't be less than a negative number.