Week #9

This week, we began to learn Big O and Big Omega and how to prove or disprove it. This part is pretty interesting, which, I think, is my favorite part in this course. The way we did the prove of Big O and Big Omega is quite different from that we did for other questions. Maybe that's why I like it:)
Although what we learnt this week is kind of easy to me, we have the second assignment that is due on Monday. Some them are pretty easy and we practice lots of time. But others seem not easy to solve like the following question.
∀ x ∈ ℝ, ∀ e ∈ ℝ!, ∃ d ∈ ℝ!, ∀ w ∈ ℝ, |x−w| < d ⇒ | x − w | < e
Here is the way I did:

    

Every time I see there are too many variables in a question, I feel confused and don't know which way should I do it first. However, there is something interesting hidden in this question, which is that this statement is actually a definition of continuous function. Since the graph of floor is obviously not a continuous function, we can say that this statement is False and we need to disprove it, which eventually gives me some ideas to solve the next question that looks similar to this one.
∃ x ∈ ℝ, ∀ e ∈ ℝ!, ∃ d ∈ ℝ!, ∀ w ∈ ℝ, |x−w| < d ⇒ | x − w | < e

Here is the way I did: