اسئله ؟؟؟

[رياضية بحتة]

Every nonempty subset of the real numbers which is bounded above has a least upper bound which is a real number.

* .if a,b  bound sets
1- show that a اتحاد b bound .
2-show that  suremump (a اتحاد b)  =suremump{sup a,supb}



           

ارجوووو منكم الرد على الموضع

[رياضية بحتة]

Theorem (supremum property: : Every nonempty set of the real numbers  that has an upper bound has asupremum in  real number?'>
2-IfA and B bound sets
a- Show that A union B dound
b- show that supremum( A union B= supremum {supremum A ,supremum B }
ارجو الحل السريع

 

[رياضية بحتة]

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Since A and B are bounded nonempty sets, then sup(A) and sup(B) exist (this follows from the least upper bound property).
Let L = max { sup(a), sup(b)
If x in (A union B), then x is either in A or in B.
(x in A) : x less than or equal sup(A) less than or equal L
(x in B) : x less than or equal sup(B) less than or equal L.
Thus L is an upper bound for (A union B).
If M is any upper bound for (A union B) then M is an upper bound for both A and B.
Since M is an upper bound for A, and sup(A) is the least upper bound for A
M is greater than or equal sup(A).
Likewise
M is greater than or equal sup(B)
But L must be equal to one or the other of sup(A), sup(B). and so M is greater than or equal L. this shows that no upper bound for (A union B) can be smaller than L.








 

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Since A is bounded, then there exists M1 in R such that
a less than or equal M1 for all a in A
Since B is bounded, then there exists M2 in R such that
b less than or equal M2 for all b in B
now let M = max (M1, M2)
for an arbitrary x in (A union B)
it is either x in A or x in B
if x in A : x less than or equal M1 less than or equal M
if x in B : x less than or equal M2 less than or equal M
hence x is less than or equal M for all x in (A union B)
(A union B ) is bounded