Question

Let A be a subset of R and suppose that s = sup(A) belongs to A. If b is not in A, show that sup(A U {b}) is equal to the larger of the two numbers s and b.

Answer 1

@wio

Answer 2

@ganeshie8

Answer 3

if b < s, show the sup is s and if the s < b, show the sup is b. b = s is trivial

Answer 4

I guess I can do the trivial case first :/
if b = s then every elements in A U {b} is less than or equal to b = s, hence sup(A U {b}) <= s = b

Answer 5

Just argue by contradiction...

Suppose \(b < s\) and \(\sup A \cup \{b\} = b\) since \(s \in A \cup \{b\}\) then by definition of supremum we have that \(s \leq b\) However this is false. So we have a contradiction

Answer 6

Now do the other case.

Answer 7

@Alchemista s does doesn't have to be in A U {b} though.

Answer 8

s *doesn't* ...

Answer 9

"Let A be a subset of R and suppose that s = sup(A) belongs to A. "

Answer 10

right. It's in A, but we don't know if it's in A U {b}

Answer 11

If it's in \(A\) then it is in \(A \cup \{b\}\)

Answer 12

:O oh I see

Answer 13

now the other case. Suppose s < b and sup(A U {b} ) = s.
for all a in A, a <= s and b <= s. (Contradiction!)

Answer 14

@Alchemista right?