Question

Show that a bounded decreasing sequence converges to its greatest lower bound.

Answer #1

Using the concept of bounded decreasing sequence I solve the problem .

If a bounded sequence is the sum of a monotone increasing and a
monotone decreasing sequence (xn = yn +
zn where {yn} is monotone increasing and {
zn} is monotone decreasing) does it follow that the
sequence converges? What if {yn} and {zn} are
bounded?

Show that every nonempty subset of the real numbers with a lower
bound has a greatest lower bound.

Show that if sequence (an) converges, then all the rearrangement
of (an) converges, and converge to the same limit

Determine whether the sequence is increasing, decreasing, or
monotonic. Is the sequence bounded?
an= 9n + 1/n

Suppose (an) is an increasing sequence of real numbers. Show, if
(an) has a bounded subsequence, then (an) converges; and (an)
diverges to infinity if and only if (an) has an unbounded
subsequence.

Problem 1 Let {an} be a decreasing and bounded
sequence. Prove that limn→∞ an exists and
equals inf{an}.

Find the least upper bound and the greatest lower bound for the
two polynomials:
a) p(x) = x4 - 3x2 - 2x + 5
b) p(x) = -2x5 + 5x4 + x3 - 3x
+ 4

Suppose (an), a sequence in a metric space X, converges to L ∈
X. Show, if σ : N → N is one-one, then the sequence (bn = aσ(n))n
also converges to L.

show that a sequence of measurable functions (fn)
converges in measure if and only if every subsequence of
(fn) has subsequence that converges in measure

Using the completeness axiom, show that every nonempty set E of
real numbers that is bounded below has a greatest lower bound
(i.e., inf E exists and is a real number).

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