Could some people help me?

[maiden beacon]

Can someone help me approach this math problem?
So, I have $u_{n+1} = \frac{1}{n+1} + \sqrt{u_n}$ a sequence and I know that $u_0 = a$ where $a \in \mathbb{R}^+$

I need to find
$\lim_{n \to \infty} (\frac{u_n - 1 - \frac{2}{n}}{\frac{1}{n}})$
How would you guys try to solve this problem?

We know that $\lim{n \to \infty} (u_n) = 1$
And that $t_n$ which is a special case of $u_n$ where $t_0 = 4$, has for limit when $n \to \infty$ the limit: $\lim{n \to \infty} (t_n) = 1$ and $\lim_{n \to \infty} (\frac{t_n - 1 - \frac{2}{n}}{\frac{1}{n}}) =0$

[hard vault]

Smile!

[spiral nexus]

my gut says you can squeeze from above and below with manipulations of t_n

I'd love to see what you studied before given this problem. I'm sure there are hints there

[spiral nexus]

$\lim{n \to \infty} (\frac{t_n - 1 - \frac{2}{n}}{\frac{1}{n}}) =0$ <-> $\lim{n \to \infty} { (n(t_n - 1)-2) =0$}

consider
$\lim{n \to \infty} { ((n \pm 1)(t_n - 1)-2)$}

[hard vault]

bodhisattv
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