12.1, due on November 28

I'm having a difficult time understanding the use of the ceiling of a number to prove what it converges to. In the proof of result 12.1, we choose a ceiling of 1/e. Since 1/e is always less than one, wouldn't the ceiling then be 1? Then we let n be an integer greater than that, or greater than one. This makes 1/n less than e. I understand all this; the reason why I'm stating it is I suppose to check my understanding. What I don't understand is how that proves that the sequence converges to 0. Or I suppose how that proves that it converges at all. Not understanding this concept makes understanding the rest of this section very tough! Unfortunately I won't be in class tomorrow due to an interview, so I won't be there for the explanation.

Not understanding that concept of the ceiling has really taken all the fun out of this section. Every proof uses that fact to prove some divergence or convergence. So I lack something specific that I enjoyed about the section. More generally it's fascinating to me that we can prove convergence in a different way than how I learned in calculus. In calculus we always ended up taking limits and using L'Hopitals rule or some other rule to show that something either converged or diverged. It's neat to have another way to show that.