Understanding Sets and Intervals: Proving Complements and Open/Closed Status

[bonildo]

Homework Statement

Hello, I'm not sure if it's the right place to post this exercise, but I'm learning it in a calculus course.

I need to prove that:

a) The complement of an open set is a closed.
b) An open interval is a open set, a closed interval is a closed set.

Homework Equations

I have the following definitions:

1) An subset A⊂R is open if for all sequence {an}n∈N that converges for l∈A,
∃n0 such that ∀n>n0 ,an∈A.

2) An subset A⊂R is closed if for all sequence {an}n∈N that converges for l∈R,
l∈A.

The Attempt at a Solution

I don't have any ideia how to do it , I never worked on this kind of exercise before

 

[haruspex]

2) An subset A⊂R is closed if for all sequence {an}n∈N that converges for l∈R,
l∈A.

That's not quite right. You need the constraint that the a_n are elements of A.
Start with an open set A and consider its complement B = R-A. Let b_n be a sequence in B converging to l in R. Consider the consequences of l not being in B. If it's not in B, where is it?