Convergence of $ s_n$ : if $s_{2n}$ increases and $s_{2n-1}$ decreases .

Question

We have a sequence $(s_n)$ such that $s_1>s_2>0$ and $s_{n+1}=\frac{1}{2}(s_n+s_{n-1})$ for $n \geq 2 $ . Show that $s_n$ converges ?

I have first observed that $s_{2n}$ increases and $(s_{2n+1})$ decreases and also proved it by finding difference between any two consective terms .

But , how shall I prove the convergence of $s_n$ and converges to what ? (in terms of $ s_1$ and $s_2$ ) .

I want to do it by using the observations which was provided as

Hint : $s_{2n}$ increases and $s_{2n+1}$ decreases .

I already proved convergence of $s_n$ but it converges to what value ?

Possible duplicate of Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy — GNUSupporter 8964民主女神地下教會, Feb 03 '19 at 10:16
I want to do it by using $s_{2n}$ increases . Not by that .see my attempt and complete that process . — Vivek, Feb 03 '19 at 10:18
But it won't hurt to see other approaches. That question is linked to many other similar questions. It will keep you busy for a while. Also, since you have 2 subsequences (increasing and decreasing) and we already revealed the limit exists, try to show the sequence is bounded. So you will have two bounded and monotonic subsequences. Then prove that these 2 subsequences have the same limit and ... you are done. — rtybase, Feb 03 '19 at 10:23
I proved that already but I want to find $s_n$ converges to what ? — Vivek, Feb 03 '19 at 10:25
In that particular case, see the linked (as a duplicate) question. One of the answers uses characteristic polynomial to find explicit/closed form for the sequence's terms. The answer depends on the initial values $s_0$ and $s_1$. — rtybase, Feb 03 '19 at 10:30
Any other method than by using characteristic polynomial since that is new to me . Any elementary method which uses the sequence $s_n$ and manipulating its terms . My professor used elementary method but i don't remember now . — Vivek, Feb 03 '19 at 10:34
Another link to the linked question suggests exploiting $s_{n+1}-s_n=\frac{s_n-s_{n-1}}{2}$ and $s_{n+1}-s_0=\sum\limits_{i} (s_{i+1}-s_{i})$. Anyway, you will find quite a few good hints. — rtybase, Feb 03 '19 at 10:40

Convergence of $ s_n$ : if $s_{2n}$ increases and $s_{2n-1}$ decreases .

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