The convergence of $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$ I would like to know if this sequence converges $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$.
I know this sequence is increasing monotone, but I couldn't prove it's bounded.
Thanks

As written it's not clear this is a sequence at all but I'm assuming your sequence is
$\sqrt{2}$, $\sqrt{2 + \sqrt{2}}$, $\sqrt{2 + \sqrt{2+ \sqrt{2}}}$, $\sqrt{2 + \sqrt{2+ \sqrt{2+\sqrt{2}}}}$, $\sqrt {2+\sqrt {2+\sqrt {2+ \sqrt{2 +\ldots}}}}$.
Then it should be obvious that each entry in the sequence is positive and the sequence increases each time.
Given this the final entry in the infinite sequence would be $x = \sqrt{2 + x}$ which we can easily solve.
$$\begin{align}
x &= \sqrt{2+x} \\
x^2 &= 2 + x\\
x^2 - x -2 &= 0\\
x &= \dfrac{1 \pm \sqrt{1+8}}{2} \\
x &= \dfrac{4}{2} = 2
\end{align}$$
The sequence has a lower bound of $\sqrt{2}$ and an upper bound of $2$ increasing monotonically.
