Question

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

Answer #1

Prove: Let S be a bounded set of real numbers and let a > 0.
Define aS = {as : s ∈ S}. Show that inf(aS) = a*inf(S).

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

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

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?

Problem 1. Let
{En}n∞=1 be a
sequence of nonempty (Lebesgue) measurable subsets of [0, 1]
satisfying
limn→∞m(En) = 1.
Show that for each ε ∈ [0, 1) there exists a subsequence
{Enk }k∞=1 of
{En}n∞=1 such
that
m(∩k∞=1Enk)
≥ ε

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

If (x_n) is a convergent sequence prove that (x_n) is bounded.
That is, show that there exists C>0 such that abs(x_n) is less
than or equal to C for all n in naturals

Prove that, for x ∈ C, when |x| < 1, lim_n→∞ |x_n| = 0.
Note:
To prove this, show that an = xn is monotone decreasing and
bounded from below. Apply the Monotone sequence theorem. Then, use
the algebra of limits, say limn→∞ |xn| = A, to prove that A =
0.

Telescoping Series. Let {an} ∞ n=0 be a sequence of real numbers
converging to zero, limn→∞ an = 0. Let bn = an − an+1. Then the
series X∞ n=0 bn converges.

Let xn be a sequence such that for every m ∈ N, m ≥ 2 the
sequence limn→∞ xmn = L. Prove or provide a counterexample: limn→∞
xn = L.

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