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

  • Thread starter Thread starter bonildo
  • Start date Start date
  • Tags Tags
    intervals Sets
Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
Sign up Log in
1 replies · 2K views
bonildo
Messages
14
Reaction score
1

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
 
on Phys.org
bonildo said:
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 an are elements of A.
Start with an open set A and consider its complement B = R-A. Let bn 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?