How to find the $\lim_{n\to\infty}s_n$ when $s_1=5, s_n =\sqrt{2+s_{n-1}}$?

Question

How to find the $\lim_{n\to\infty}s_n$ when $s_1=5, s_n =\sqrt{2+s_{n-1}}$ using the Monotone Convergence Theorem?

I have the proof from my professor, but I am stuck at one step.

Proof:

Step 1: Show that it is monotonic:

Proof by induction: Claim: $s_n > s_{n+1}, \forall n\in\mathbb{N}$

Base case: $n=1$. $s_1 = 5 > s_2=\sqrt{7}$, so it holds for $n=1$.

Assume it holds for $n=k$, now show it's true for $n=k+1$.

Assume $s_k>s_{k+1}$, then $s_{(k+1)+1} = s_{k+2} = \sqrt{2+s_{k+1}} < \sqrt{2+s_k} = s_{k+1}$ by assumption. Therefore, by the Principle of Mathematical Induction, the statement is true $\forall n\in\mathbb{N}$ and $\{S_n\}$ is decreasing (monotonic).

Step 2: Find the limit:

It is clear $0\leq s_n\leq s_1 = 5\forall n$, so $\{S_n\}$ is bounded.

Therefore, since $\{S_n\}$ is monotone and bounded, by the Monotone Convergence Theorem, $\{S_n\}$ converges.

Let $\lim_{n\to\infty} s_n = L$: \begin{align} \lim_{n\to\infty}\sqrt{2+s_{n-1}} &= L\\ \sqrt{2 + \lim_{n\to\infty} s_{n-1}} &= L\\ \sqrt{2+L} &= L *\text{This is where I get stuck...}\\ 2+L &= L^2\\ 0 &= L^2 - L - 2\\ &= (L-2)(L+1)\\ &\implies L=-1, 2 \end{align} But $L\ne-1$ since $s_n\geq 0\forall n$. Therefore, $\lim_{n\to\infty} s_n = 2$.

I get stuck because I am unsure why we can say $\lim_{n\to\infty} s_{n-1} = L$. We said that $\lim_{n\to\infty} s_{n} = L$, but nothing about the limit of $s_{n-1}$.

Answer 1

Write down the definition of a limit to show that if $(a_n)$ is any converging sequence, then $(a_{n-1})$ is also a converging sequence and both limits are the same.

Answer 2

The sequence is decreasing, because $s_{n+1}<s_n$ is equivalent to $$ s_n+2<s_n^2 $$ which is satisfied when $s_n<-1$ or $s_n>2$. The latter is true by induction: $s_1=5>2$; suppose $s_k>2$; then $s_{k+1}=\sqrt{s_k+2}>\sqrt{2+2}=2$.

Your proof is good as well.

The sequence is bounded below by $0$, hence it converges. If $l$ is its limit, then $l\ge0$ and $$ l=\lim_{n\to\infty}s_n=\lim_{n\to\infty}s_{n+1}=\sqrt{l+2} $$ Therefore $l=2$.

Actually, $\lim_{n\to\infty}s_{n+1}$ should really be $\lim_{n\to\infty}s'_n$, where $$ s'_n=s_{n+1} $$ It's essentially going through the definition of limit to show that $(s_n)$ converges if and only if $(s'_n)$ and, in this case, they have the same limit.

Since $s'_n=\sqrt{s_n+1}$, the theorems on limits also allow us to say that $$l=\lim_{n\to\infty}s'_n=\lim_{n\to\infty}\sqrt{s_n+2}=\sqrt{l+2}$$

Note: I wouldn't use $s_{n-1}$, but it's just personal preference.