Question

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

Answer #1

12

Prove that every bounded sequence has a convergent
subsequence.

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

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

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.

Find a sequence of positive Lebesgue integrable functions on
[0,1] which do not converge pointwise (it means that there is no
point x0 so that fn(x0 ) is a
convergent sequence) but its integrals do converge to zero.

4.2.7. Example. If (xn) is a sequence in (0, ∞) and xn → a,
then √xn → √a.
?
Proof. Problem. Hint. There are two possibilities: treat the
cases a = 0 and a > 0 sepa- √√
rately. For the first use problem 4.1.7(a). For the second use
4.2.1(b) and 4.1.11; write xn − a as |xn − a|/(√xn + √a). Then find
an inequality that allows you to use the sandwich theo-
rem(proposition 4.2.5).

Determine whether the following sequences converge or diverge.
If a sequence converges, find its limit. If a sequence diverges,
explain why.
(a) an = ((-1)nn)/
(n+sqrt(n))
(b) an = (sin(3n))/(1- sqrt(n))

Prove that the sequence cos(nπ/3) does not converge.
let epsilon>0
find a N so that |An| < epsilon for n>N

find the nth term of the following sequence, does it
converge?
{2/1, 6/10, 12/100, 60/1000....}

construct an example of sequence of functions that the family of
such function is uniformly bounded but does not have subsequence
that converges uniformly, with detail proof.

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