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1. We dene a sequence (an), n in N recursively bya_1 = -2 and a_(n+1) = -sqrt( 4 -a_n) Note that (an) n in N is well-defined and we have an < 0 for all n in NA. Show that (an), n in N is decreasing and bounded below.Hint: Show that -3 < an+1 < an, for all n in N by induction.B. Show that (an) n in N converges. Moreover, compute lim n->oo. You can use part 1, even if you did not solve it.2.Let (an) n in N be a bounded sequence. We set L = lim sup( an); l = lim inf (an):A. Show that L; l belong to R.B. Show that for all ϵ > 0, there exists n0 belong to N such that for all n > n0, we have l – ϵ < an < L + ϵ: You can use part 1, even if you did not solve it. Hint: Use the definition of lim sup; sup; lim inf; inf.3.Let (an) n in N be a sequence such that lim n->oo (an) = 0: Show that for every ϵ > 0, there exists a subsequence (an(k)) k belong to N such that the series Σ1 {k=1,oo} an(k) converges and |Σ1{k=1,oo} an(k)| < ϵ: Hint: Show that there exists 0 < r < 1 such that Σ1{n=1,oo} rn = r/(1-r) < ϵ: Then construct appropriate (n(k)) k belong to N inductively.4.Let (an) n in N, (bn) n in N be sequences such that |an+1 - an| < bnfor all n in N and the series Σ1{n=1,oo} bn converges. Show that THE SEQUENCE (an) n in N converges.5.Dene a function f : R {0} -> R by f(x) =( x^2 – 1)/xShow that f is continuous on the domain using the ϵ- g denition of continuity. For this problem, it is forbidden to use results on continuity from the textbook and lectures.

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Problem 1 (10 points). We define a sequence (an )n∈N recursively by

a1 = −2

and

√

an+1 = − 4 − an (n ∈ N).

Note that (an )n∈N is well-defined and we have an < 0 for all n ∈ N.
1. (6 points) Show that (an )n∈N is decreasing and bounded below.
Hint: Show that
−3 < an+1 < an
for all n ∈ N by induction.
2. (4 points) Show that (an )n∈N converges. Moreover, compute limn→∞ an . You can use
part 1, even if you did not solve it.
Problem 2 (10 points). Let (an )n∈N be a bounded sequence. We set
L = lim sup an , l = lim inf an .
1. (4 points) Show that L, l ∈ R.
2. (6 points) Show that for all ϵ > 0, there exists n0 ∈ N such that for all n ≥ n0 , we have

l − ϵ < an < L + ϵ.
You can use part 1, even if you did not solve it.
Hint: Use the definitions of lim sup, sup, lim inf, inf.
Problem 3 (10 points). Let (an )n∈N be a sequence such that
lim an = 0.
n→∞
Show that for every ϵ > 0, there exists a subsequence (an(k) )k∈N such that the series

∞

∑

an(k)

k=1

converges and

∞

∑

an(k) < ϵ.
k=1
Hint: Show that there exists 0 < r < 1 such that
∞
∑
n=1
rn =
r
< ϵ.
1−r
Then construct appropriate (n(k))k∈N inductively.
Problem 4 (10 points). Let (an )n∈N , (bn )n∈N be sequences such that
|an+1 − an | < bn
for all n ∈ N and the series
∞
∑
bn
n=1
converges. Show that THE SEQUENCE (an )n∈N converges.
Problem 5 (10 points). Define a function f : R {0} → R by
f (x) =
x2 − 1
.
x
Show that f is continuous on the domain using the ϵ-δ definition of continuity. For this
problem, it is forbidden to use results on continuity from the textbook and
lectures.
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Tags:
dene a sequence
bounded sequence
belong to R
definition of lim sup
denition of continuity
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