I understand the definition of a convex function to be the following:
$$ \alpha f(x) + (1 - \alpha) f(y) \geq f(\alpha x + (1-\alpha) y)$$
Intuitively, this means that any point on a line connecting some $x$ and $y$ on the function should exist above a corresponding point on the function. What is confusing me is that for some proofs, all that is shown is that the right hand subtracted from the left hand is greater than or equal to zero. It makes sense because one thing minus another greater than zero implies the first thing must be greater than the second. I guess what confuses me is that I cant understand when it would be the case that subtracting right from left wouldn't produce some difference that is greater than/equal to zero. Sorry if this question is poorly worded. I will try to clarify what I mean if anybody asks.