combining big Oh with bit Theta

Hi guys : I've noticed that in some proofs, they combine big Oh notation with big Theta.... I'm not sure, in general, why you would want to express the Upper bound (big Oh) of an algorithm in terms of the lower bound (big theta).....

Any thoughts on the combination of big Oh, horseshoe Oh, and Theta Oh in algorithm analysis ?

Theta is not lower bound. Omega (or horseshoe, as you call it) is the lower bound. If an algorithm is in Theta for some function, it means that the lower bounds and the upper bound are both described by that function as well.

For instance, if an algorithm is in O(n log n), and it is in Omega(n log n) as well, we say the algorithm is in Theta(n log n).

Hmmm okay... So I guess a "tight" bound is when omega=theta ?

A bound g(n) is tight if it's in O(f(n)) and Omega(f(n)) and thus in Theta(f(n)).