The question along with its solution are attached.

I'm stuck on the step where it compares the sum to the integral with the <= sign. I don't understand how someone can infer the truth of this in a way that requires no additional steps.

Basically, how can I show this to be true without requiring too much work? (I just want to grasp it intuitively - I don't need a full proof or anything).

Any help would be greatly appreciated!

Thanks in advance!