Question

Show that sequence {s_{n}} converges if it is monotone
and has a convergent subsequence.

Answer #1

Suppose that every Cauchy sequence of X has a convergent
subsequence in X. Show that X is complete.

Prove that every bounded sequence has a convergent
subsequence.

Find an example of a sequence, {xn}, that does
not converge, but has a convergent subsequence.
Explain why {xn} (the divergent sequence) must have an
infinite number of convergent subsequences.

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.

Prove that if a sequence converges to a limit x then very
subsequence converges to x.

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

suppose that the sequence (sn) converges to s. prove that if s
> 0 and sn >= 0 for all n, then the sequence (sqrt(sn))
converges to sqrt(s)

Prove: If x is a sequence of real numbers that converges to L,
then any subsequence of x converges to L.

Prove that X is totally bounded if every sequence of X has a
convergent subsequence. Please directly prove it without using any
theorem on totally boundedness.

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?

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