Finding the number of iterations to a recurrence
Presumably the correct form of the recursive step is
$$ S_{k+1} \leq S_k - S_k \frac{\sigma}{1+\sigma} = \frac{S_k}{1+\sigma}. $$
Using induction, we can show that
$$ S_t \leq \frac{S}{(1+\sigma)^t}. $$
So the algorithm will end by the time $T$ satisfying
$$ \frac{S}{(1+\sigma)^T} &lt; 1.$$
I'll let you solve this yourself.