Help two questions in math. analysis .

[ShengyaoLiang]

help...two questions in math. analysis...

1) prove : every bounded open interval(a,b) where a less than b and a,b in R , then is an open set.

2) If c = supA and a is not the maximum of A, then sup(A\{a})= c.

I really don't know how to prove those two...thanks for helping me...really...

 

[SiddharthM]

if x is in (a,b) then a<x<b, so what can you say about b-x and x-a? Choose a epsilon neighborhood of x that is contained in (a,b) appropriately.

we have that c=supA and that a is not the max of A. The latter statement needs some unraveling - I presume a IS inside A so then it means that a is NOT an upper bound of A, i.e. there exists a b in A s.t. b>a. Take epsilon to be small and positive, then look at c minus epsilon. Remember there is a b in A with b>a, which also means that b IS not a.

 

[Sourabh N]

Do you know what is an open set ?

 

[ShengyaoLiang]

if x is contained in (a,b) and there is no intersection with ]a,b[ , then it is an open set?

 

[Sourabh N]

A better way to define open set would be to say every point in the neighborhood is in the set.

 

[Sourabh N]

Just read the reply by SidharthM, and read about open interval, open set, etc. on wikipedia. Giving the whole solution would not help in your understanding.