Comment by otasevic

thanks for this comment. I realized that I skipped a few steps in the explanation, so I am updating it now.

But essentially you see that the equation is S^2 - S - 2L < 0

from that, you can solve that the roots of the function are: (1) 1/2 + sqrt(2L) and (2) 1/2 - sqrt(2L)

That means that (S-1/2-sqrt(2L)) * (S-1/2+sqrt(2L)) < 0

The second term is always positive, which means that we need to make the first term negative in order for the inequality to hold.

=> S < 1/2 + sqrt(2L) which leads to O(sqrt(L)) when you let L approach infinity.

Does this make sense?