$s_3 = 1 + \min(s_1, s_2) \\= 1 + \min(1, 1) = 2 = \lceil \frac{3}{2} \rceil$.
So, base condition of induction satisfied.
Assume, $s_{n-2} =\lceil \frac{n-2}{2} \rceil$ and $s_{n-1} =\lceil \frac{n-1}{2} \rceil$ (Induction hypothesis)
Now, we have to prove,
$s_n = \lceil \frac{n}{2} \rceil$
$s_n = 1 + \min(s_{n-1}, s_{n-2}) \\= 1 + \lceil \frac{n-2}{2} \rceil \\= 1 + \lceil \frac{n}{2} \rceil -1 \\=\lceil \frac{n}{2}\rceil$
(Hence, proved)