LECTURE
2
SEQUENCES OF REAL NUMBERS
Definition 2.1 By a sequence of real numbers (or sequence in R) we mean any function of type
f : N → R. Denoting xn ..= f (n) for all n ∈ N we may use one of the following notations to represent
the sequence f :
(xn )n∈N , (xn )n≥1 or simply (xn ).
Remark 2.2 Any function g : N ∩ [m, ∞) → R with m ∈ Z can be seen as a sequence, too. Indeed,
g can be identified with f : N → R, given by
f (n) ..= g(m + n − 1), ∀ n ∈ N.
In this case instead of (xn )n∈N , with xn ..= f (n) for all n ∈ N, we write
(yn )n≥m , where yn ..= g(n), ∀ n ∈ Z, n ≥ m.
Definition 2.3 A sequence of real numbers (xn )n∈N is said to be bounded below (bounded above;
bounded; unbounded) if the set {xn | n ∈ N} is bounded below (bounded above; bounded; unbounded).
Remark 2.4 For any sequence (xn )n∈N in R we have:
(xn )n∈N is bounded below
(xn )n∈N is bounded above
(xn )n∈N is bounded
(xn )n∈N is unbounded
⇐⇒
⇐⇒
⇐⇒
⇐⇒
∃ a ∈ R s.t. xn ≥ a, ∀ n ∈ N;
∃ a ∈ R s.t. xn ≤ a, ∀ n ∈ N;
∃ a ∈ R s.t. |xn | ≤ a, ∀ n ∈ N;
∀ a ∈ R, ∃ n ∈ N s.t. |xn | > a.
Definition 2.5 A sequence (xn )n∈N in R is called
• increasing if xn+1 ≥ xn , ∀n ∈ N;
• decreasing if xn+1 ≤ xn , ∀n ∈ N;
• strictly increasing if xn+1 > xn , ∀n ∈ N;
• strictly decreasing if xn+1 < xn , ∀n ∈ N;
• monotonic (or monotone) if it is increasing or decreasing;
• strictly monotonic (or strictly monotone) if it is strictly increasing or strictly decreasing.
Remark 2.6 Every increasing sequence is bounded below. Similarly, every decreasing sequence is
bounded above.
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Definition 2.7 We say that a sequence of real numbers (xn )n∈N has a limit (in R) if there exists
` ∈ R ..= R ∪ {−∞; +∞} such that
∀ V ∈ V(`), ∃ nV ∈ N s.t. xn ∈ V, ∀ n ∈ N, n ≥ nV .
(2.1)
Remark 2.8 For any `1 , `2 ∈ R, `1 6= `2 , there exist V1 ∈ V(`1 ) and V2 ∈ V(`2 ) s.t. V1 ∩ V2 = ∅.
Hence, whenever (xn )n∈N has a limit, there is a unique ` ∈ R satisfying (2.1).
Definition 2.9 If a sequence (xn )n∈N has a limit, then the unique ` ∈ R that satisfies (2.1) is called
the limit of the sequence (xn )n∈N . In this case we denote lim xn ..= ` or xn → ` and we say that
n→∞
(xn )n∈N tends to `.
Proposition 2.10 Let (xn )n∈N be a sequence of real numbers that has a limit. We have
lim xn = ` ∈ R ⇐⇒ ∀ ε > 0, ∃ nε ∈ N s.t. |xn − `| < ε, ∀ n ∈ N, n ≥ nε ;
n→∞
lim xn = −∞ ⇐⇒ ∀ a ∈ R, ∃ na ∈ N s.t. xn < a, ∀ n ∈ N, n ≥ na ;
n→∞
lim xn = +∞ ⇐⇒ ∀ a ∈ R, ∃ na ∈ N s.t. xn > a, ∀ n ∈ N, n ≥ na .
n→∞
Definition 2.11 A sequence of real numbers (xn )n∈N is said to be
• convergent, if it has a finite limit, i.e., lim xn ∈ R; in this case we say that (xn )n∈N converges
n→∞
to the real number lim xn ;
n→∞
• divergent, if it is not convergent.
Proposition 2.12 Let (xn )n∈N be a sequence of real numbers and let ` ∈ R. The following assertions
are equivalent:
1◦ The sequence (xn )n∈N converges to `, i.e., lim xn = `.
n→∞
2◦ The sequence (|xn − `|)n∈N converges to zero, i.e., lim |xn − `| = 0.
n→∞
3◦ There exists a sequence (an )n∈N of nonnegative real numbers satisfying the following two conditions:
(i) |xn − `| ≤ an for all n ∈ N;
(ii) (an )n∈N converges to zero, i.e., lim an = 0.
n→∞
Corollary 2.13 Let (an )n∈N and (bn )n∈N be two sequences of real numbers. If (an )n∈N is bounded
and lim bn = 0, then lim (an · bn ) = 0.
n→∞
n→∞
Proposition 2.14 Let (xn ), (yn ) be sequences in R such that ∀n ∈ N, xn ≤ yn .
(i) If (xn ) and (yn ) converge, then lim xn ≤ lim yn .
n→∞
n→∞
(ii) If lim xn = +∞, then lim yn = +∞.
n→∞
n→∞
(iii) If lim yn = −∞, then lim xn = −∞.
n→∞
n→∞
Theorem 2.15 (Squeeze Theorem) Let (xn ), (yn ), (zn ) be sequences in R such that
xn ≤ yn ≤ zn , ∀ n ∈ N.
Suppose that (xn ) and (zn ) are convergent and lim xn = lim zn = ` ∈ R. Then (yn ) is also
n→∞
n→∞
convergent and lim yn = `.
n→∞
Corollary 2.16 (Cantor’s Theorem on Nested Intervals) Consider a nested sequence of compact intervals, i.e.,
In ..= [an , bn ] ⊆ R, s.t. an ≤ an+1 < bn+1 ≤ bn , ∀ n ∈ N.
If lim (bn − an ) = 0, then there exists x ∈ R such that
n→∞
∞
\
n=1
2
In = {x}.
Proposition 2.17 Every convergent sequence of real numbers is bounded.
Notice that there are bounded sequences that are not convergent.
Theorem 2.18 (Counterpart of the Weierstrass’ Theorem) Let (xn )n∈N be a monotonic (i.e.,
increasing or decreasing) sequence of real numbers. The following assertions hold:
1◦ (xn )n∈N has a limit in R.
2◦ If (xn )n∈N is increasing, then lim xn = sup xn , hence
n→∞
n∈N
(xn ) is convergent if and only if it is bounded above.
2◦ If (xn )n∈N is decreasing, then lim xn = inf xn , hence
n→∞
n∈N
(xn ) is convergent if and only if it is bounded below.
Theorem 2.19 (Toeplitz) Consider an ”infinite triangular matrix” of real numbers
c1,1
c2,1
···
cn,1
···
c2,2
··· ···
cn,2 cn,3 · · · cn,n
··· ··· ··· ···
which satisfies the following three conditions:
(i) cn,k ≥ 0, ∀ n ∈ N, ∀ k ∈ {1, 2, ..., n};
n
P
(ii)
cn,k = 1, ∀n ∈ N;
k=1
(iii)
lim cn,k → 0, ∀ k ∈ N.
n→∞
If (xn )n∈N is a sequence of real numbers that has a limit ` ∈ R then the sequence (yn )n∈N , given by
yn = cn,1 x1 + cn,2 x2 + ... + cn,n xn , ∀ n ∈ N
has the same limit `.
Theorem 2.20 (Stolz-Cesàro) Let (an ), (bn ) be sequences in R such that
(i) (bn ) is strictly increasing and lim bn = +∞,
n→∞
an+1 − an
= L ∈ R.
(ii) lim
n→∞ bn+1 − bn
an
Then lim
= L.
n→∞ bn
Proof. We can apply Toeplitz’s Theorem, letting a0 = b0 ..= 0,
xn ..=
Since yn =
an
,
bn
an − an−1
, ∀n ∈ N
bn − bn−1
and
cn,k ..=
bk − bk−1
, ∀ k ∈ {1, 2, ..., n}.
bn
the conclusion follows.
Corollary 2.21 Let (xn ) be a sequence in R. The following assertions hold:
x1 + x2 + . . . + xn
1◦ If lim xn = ` ∈ R, then lim
= `.
n→∞
n→∞
n
√
2◦ If xn > 0 for all n ∈ N and lim xn = ` ∈ R, then lim n x1 · x2 · . . . · xn = `.
n→∞
n→∞
√
x
n+1
3◦ If xn > 0 for all n ∈ N and lim
= L ∈ R, then lim n xn = L.
n→∞ xn
n→∞
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Definition 2.22 Let (xn )n∈N be a sequences of real numbers. A sequence (yk )k∈N is said to be a
subsequence of (xn )n∈N if there exists a strictly increasing sequence (nk )k∈N of natural numbers (i.e.,
nk ∈ N and nk < nk+1 for all k ∈ N) such that
yk = xnk , ∀ k ∈ N.
Proposition 2.23 Let (xn )n∈N be a sequence in R that has a limit x = lim xn ∈ R. Then any
n→∞
subsequence (xnk )k∈N of (xn )n∈N has the same limit, i.e., lim xnk = x.
k→∞
Theorem 2.24 (Bolzano-Weierstrass) Every bounded sequence of real numbers has a convergent
subsequence.
Theorem 2.25 (Cauchy’s criterion for convergence of sequences) For any sequence (xn )n∈N
the following assertions are equivalent:
1◦ (xn )n∈N is convergent.
2◦ For every ε > 0 there exists nε ∈ N such that for all m, n ∈ N, with m ≥ nε and n ≥ nε , we
have |xm − xn | < ε.
3◦ For every ε > 0 there exists nε ∈ N such that for any n ∈ N with n ≥ nε and any p ∈ N we
have |xn+p − xn | < ε.
Corollary 2.26 (Sufficient condition for convergence of sequences) Let (xn )n∈N be a sequence
of real numbers. Assume that there is a sequence (an )n∈N of nonnegative real numbers satisfying the
following two conditions:
1◦ |xn+p − xn | ≤ an for all n, p ∈ N;
2◦ (an )n∈N converges to zero, i.e., limn→∞ an = 0.
Then the sequence (xn )n∈N is convergent.
Limit Laws
x + ∞ = ∞ + x = ∞, ∀x ∈ R,
x + (−∞) = (−∞) + x = −∞, ∀x ∈ R,
∞ + ∞ = ∞, (−∞) + (−∞) = −∞,
∞, if x ∈ (0, ∞)
x·∞=∞·x=
−∞, if x ∈ (−∞, 0),
−∞, if x ∈ (0, ∞)
x · (−∞) = (−∞) · x =
∞, if x ∈ (−∞, 0),
∞ · ∞ = ∞, (−∞) · (−∞) = ∞, ∞ · (−∞) = (−∞) · ∞ = −∞,
x
x
= −∞
= 0, ∀x ∈ R,
∞
1
0+
1
= ∞, 0−
= −∞,
∞, if x ∈ (1, ∞)
x∞ =
0, if x ∈ [0, 1),
0, if x ∈ (1, ∞)
x−∞ =
∞, if x ∈ (0, 1),
∞, if x ∈ (0, ∞)
(∞)x =
0, if x ∈ (−∞, 0),
∞∞ = ∞, ∞−∞ = 0.
The following limits are undefined
∞ + (−∞), (−∞) + ∞,
0 · ∞, ∞ · 0, 0 · (−∞), (−∞) · 0,
∞
∞
, −∞
, −∞
, −∞
,
∞
−∞
∞
1∞ ,
00 ,
∞0 ,
1−∞ .
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