1. hw 4
Remark: All the sequences are sequence of real numbers.
Formula that might be useful: Let θ ∈ R.
a. sin 2θ = 2 sin θ cos θ
b. cos 2θ = cos2θ − sin2θ = 2 cos2θ − 1 = 1 − 2 sin2θ,
(1) Let (xn) be the sequence defined by xn= 1 +1
2 + · · · + 1
n, n ≥ 1.
Prove that (xn) is divergent to ∞, in other words, lim
n→∞xn= ∞.
(2) Let S = {x ∈ R : x2+ x < 3}.
(a) Show that S is nonempty and bounded.
(b) Find sup S and inf S. Here you need to verify the results you obtained.
(3) Let S be a nonempty set. Suppose S is bounded above. Show that there exists a sequence (xn) with xn ∈ S so that lim
n→∞xn = sup S. (Similarly, if S is bounded below, you can show that there exists a sequence (yn) with yn∈ S so that lim
n→∞yn= inf S. )
(4) Find a bounded sequence that has three cluster points.
(5) Find limsup and liminf of (an).
(a) an= 3 + (−1)n
1 +1
n
, n ≥ 1.
(b) an= (−1)n
n +1 + (−1)n
2 , n ≥ 1.
(c) an= 1 + n
n + 1cosnπ
2 , n ≥ 1.
(6) Let (an) and (bn) be bounded sequences. Show that lim sup
n→∞
(an+ bn) ≤ lim sup
n→∞
an+ lim sup
n→∞
bn.
(7) Let (an) be a sequence convergent to a ∈ R. Let (σn) be a sequence defined by σn= a1+ a2+ · · · + an
n , n ≥ 1.
Show that lim
n→∞σn= a.
(8) Let 0 < θ < π/2. Define a sequence (an) by an= cosθ
2cos θ
22 · · · cos θ 2n.
Show that (an) is convergent and find its limit. (Hint: use sin 2θ = 2 sin θ cos θ )
1
2
(9) Suppose 0 < a1 < b1. Denote cos θ = a1 b1
. Define two sequences (an) and (bn) as follows.
an+1 = an+ bn
2 , bn+1=p
an+1bn, n ≥ 1.
(a) Show that the inequality
an< an+1< bn+1< bn for all n ≥ 1.
(b) Show that lim
n→∞an = lim
n→∞bn = sin θ
θ b1. (Hint: cos 2θ = cos2θ − sin2θ = 2 cos2θ − 1 = 1 − 2 sin2θ, )
