Can't find where my mistake is: Majorization of a sequence

Question

I'm working on the following sequence $$ u_{n+1} = \sqrt{n+u_n} \text{ with } u_1=1 $$ I've shown that $$ u_n \leq n $$ Then $$ u_{n+1} \leq \sqrt{2}\sqrt{n} \Leftrightarrow \frac{u_{n+1}}{\sqrt{n}} \leq \sqrt{2} $$ which I've verified with a python script that it looks true. But then $$ u_n = \left(u_{n+1}\right)^2 - n = \left(u_{n+1} - \sqrt{n}\right)\left(u_{n+1}+\sqrt{n}\right) $$ Hence $$ \frac{u_n}{\sqrt{n}} = \left(\frac{u_{n+1}}{\sqrt{n}} - 1\right)\left(\frac{u_{n+1}}{\sqrt{n}} + 1\right) $$ Using the result above, then $$ \frac{u_n}{\sqrt{n}} \leq \left(\sqrt{2} - 1\right)\left(\sqrt{2}+1\right) = 1 $$ It suggests that $u_n \leq \sqrt{n}$ for all $n \geq 1$ however this is not true and I can't seem to find what mistakes I've done. I'm sure it's just a "stupid" mistake that I can't see somehow, any help would be great.

Answer 1

$\dfrac{ab}n=\dfrac{a}{\sqrt{n}}\times\dfrac b{\sqrt{n}}$

You actually divided by $n$ not $\sqrt{n}$ so you just get back $\dfrac{u_n}n\le 1$ you know already.