A community for students. Sign up today!
Here's the question you clicked on:
 0 viewing
 one year ago
Show that any nonempty finite set S ⊂ ℝ contains both its supremum and infimum. (Hint: use induction)
 one year ago
Show that any nonempty finite set S ⊂ ℝ contains both its supremum and infimum. (Hint: use induction)

This Question is Closed

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0To be honest I really didnt study the chapter fully yet I was just looking for a tutor so hopefully I wont be lost lol

abb0t
 one year ago
Best ResponseYou've already chosen the best response.1\[S \subseteq R\] such that S = n Proceed by induction on n. If n = 1, then S = {a} for some \[a \in R\]. Then, sup S = a. Assume that all subsets of R of order k contain their respective suprema. Let S be a subset of R in order of k+1. Label the elements of S as: \[\left\{ a_1, a_2...a_{k+1} \right\}\] Let \[T = \left\{ a_1, a_2...a_k \right\} \] By the inductive assumption: \[\sup T= a_{i_o}\] for some \[1 \le i_o \le k\] and \[\sup(ST) = a_{k+1}\] Thus clamining case 1: \[a_{i_o} \le a_{k+1}\] since \[a_{i_o} = \sup T\] and \[a_{i_o} \le a_{k+1}\], you know that for all \[1 \le i_o \le k, a_i \le a_{i_o} \le a_{k+1}\] thus \[\sup S = a_{k+1}\] Case 2: \[a_{i_o} \ge a_{k+1}\] since \[a_{i_o} = \sup T\] and \[a_{i_o} \ge a_{k+1} \] you know that for all \[1 \le i \le k+1, a_i \le a_{i_o}\] thus \[\sup S = a_{i_o}\] Thus, S contains its supremum

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0Give me time to read it. Thanks :)

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0Wow this is great :)

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0I didnt follow case 2 though

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0Ummm Also werent we supposed to prove that it contains its supremum and infimum

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0Like I followed what u did but i am just confused why we needed to show case 2

abb0t
 one year ago
Best ResponseYou've already chosen the best response.1Well supremem is the upper bound, and infimum is the lower bound. I just used the definitions.

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0No like how did u show S contains the infimum

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0Ohhhhhh ok I thought i was going senile

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0lol Thankssssssss. THIS WAS AWESOME. U R REALLY CLEARRRRRRRR
Ask your own question
Ask a QuestionFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.