Question

Let (sn) ⊂ (0, +∞) be a sequence of real numbers. Prove that

liminf 1/Sn = 1 / limsup Sn

Answer #1

Let (sn) be a sequence. Consider the set X consisting of real
numbers x∈R having the following property: There exists N∈N s.t.
for all n > N, sn< x. Prove that limsupsn= infX.

Claim: If (sn) is any sequence of real numbers with
??+1 = ??2 + 3?? for
all n in N, then ?? ≥ 0 for all n in N.
Proof: Suppose (sn) is any sequence of real numbers
with ??+1 = ??2 + 3??
for all n in N. Let P(n) be the inequality statements ??
≥ 0.
Let k be in N and suppose P(k) is true: Suppose ?? ≥
0.
Note that ??+1 = ??2 +
3?? =...

Do not use binomial theorem for this!! (Real analysis
question)
a) Let (sn) be the sequence deﬁned by sn = (1 +1/n)^(n). Prove
that sn is an increasing sequence with sn < 3 for all n.
Conclude that (sn) is convergent. The limit of (sn) is referred to
as e and is used as the base for natural logarithms.
b)Use the result above to ﬁnd the limit of the sequences: sn =
(1 +1/n)^(2n)
c)sn = (1+1/n)^(n-1)

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)

Let
<Xn> be a cauchy sequence of real numbers. Prove that
<Xn> has a limit.

Prove that if (xn) is a sequence of real numbers,
then lim sup|xn| = 0 as n approaches infinity. if and
only if the limit of (xN) exists and xn
approaches 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.

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).

Given that xn is a sequence of real numbers. If (xn) is a
convergent sequence prove that (xn) is bounded. That is, show that
there exists C > 0 such that |xn| less than or equal to C for
all n in N.

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.

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