Question

let A be a nonempty subset of R that is bounded below. Prove that inf A = -sup{-a: a in A}

Answer #1

i hope you like it... this is very famous question of real analysis.. thanks dear

Suppose A ⊆ R is nonempty and bounded above and β ∈ R. Let A + β
= {a + β : a ∈ A}
Prove that A + β has a supremum and sup(A + β) = sup(A) + β.

Let A⊆R be a nonempty set, which is bounded above. Let
B={a-5:a∈ A}. Prove that sup(B)=sup(A)-5

Let S and T be nonempty subsets of R with the following
property: s ≤ t for all s ∈ S and t ∈ T.
(a) Show that S is bounded above and T is bounded below.
(b) Prove supS ≤ inf T .
(c) Given an example of such sets S and T where S ∩ T is
nonempty.
(d) Give an example of sets S and T where supS = infT and S ∩T
is the empty set....

Prove that if E ⊂ R is bounded and sup E ∈ E or inf E ∈ E, then
E is not open.
Analysis question (R refers to the real numbers).

Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤ (a1)2 + (a2)2 +
... + (an)2.
Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).
Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s...

Let f : [a,b] → R be a bounded function and let:
M = sup f(x)
m = inf f(x)
M* =sup |f(x)|
m* =inf |f(x)|
assuming you are taking values of x that lie in
[a,b].
Is it true that M* - m* ≤ M - m ?
If it is true, prove it. If it is false, find a counter
example.

Prove:
A nonempty subset C⊆R is closed if and only if
there is a continuous function g:R→R such that
C=g-1(0).

Let A be open and nonempty and f : A → R. Prove that f is
continuous at a if and only if f is both upper and lower
semicontinuous at a.

1. (a) Let S be a nonempty set of real numbers that is bounded
above. Prove that if u and v are both least upper bounds of S, then
u = v.
(b) Let a > 0 be a real number. Deﬁne S := {1 − a n : n ∈ N}.
Prove that if epsilon > 0, then there is an element x ∈ S such
that x > 1−epsilon.

Suppose S and T are nonempty sets of real numbers such that for
each x ∈ s and y ∈ T we have x<y.
a) Prove that sup S and int T exist
b) Let M = sup S and N= inf T. Prove that M<=N

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 12 minutes ago

asked 13 minutes ago

asked 23 minutes ago

asked 48 minutes ago

asked 51 minutes ago

asked 52 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago