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Show that any nonempty finite set S ⊂ ℝ contains both its supremum and infimum. (Hint: use induction)
 one year ago
 one year ago
Show that any nonempty finite set S ⊂ ℝ contains both its supremum and infimum. (Hint: use induction)
 one year ago
 one year ago

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

abb0tBest 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
 one year ago

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

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

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

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

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

abb0tBest ResponseYou've already chosen the best response.1
Well supremem is the upper bound, and infimum is the lower bound. I just used the definitions.
 one year ago

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

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

swissgirlBest ResponseYou've already chosen the best response.0
lol Thankssssssss. THIS WAS AWESOME. U R REALLY CLEARRRRRRRR
 one year ago
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