I try to determine if $ f (n) = sqrt n $ is in $ O (g (n)) $, $ Omega (g (n)) $, or $ Theta (g (n)) $ from where $ g (n) = log ^ 2n $,

The answer says only that $ f (n) = theta (g (n)) $ is right, but why not $ f (n) = O (g (n)) $ right, too?

The formal definition is $ f (n) = O (g (n)) $ means $ c * g (n) $ is an upper bound on $ f (n) $, So there is a certain constant c, that $ f (n) $ is always $ leq $ $ c * g (n) $ for big enough $ n $,

If we take n = 100,000 (I think that's big enough), then

$$ sqrt {100,000} leq c * log ^ 2100,000 $$

$$ sim 316 leq c * 25 $$

Here we actually see one $ c $ such as $ c = 15 $ that will satisfy the inequality. That makes no sense, because then it should be according to this logic $ c $ that will satisfy any inequality. Do I interpret the definitions incorrectly?