Fundamentals of Pure Mathematics 2023-24
Analysis workshop in week 2
Sequences
The cornerstone of Analysis is the Theory of Limits. Derivatives are limits
of difference quotients, definite integrals are limits of Riemann sums, infinite sums are limits of partial sums, ex is the limit of ern where (rn ) is a
sequence of rationals converging to the real number x, and so on. We’ve
started our study of limits by looking at convergent sequences of real numbers.
A property Pn involving a positive integer n (e.g., 2n > 5n2 ) is said to be
eventually true iff there exists a positive integer N such that for all positive
integers n with n ≥ N, Pn is true. For example, 2n > 5n2 is eventually true,
because it is true for all n with n ≥ 1000 (there are n’s smaller than 1000
for which it is true and there are n’s smaller than 1000 for which it is false).
Some authors prefer the expression ‘Pn is true for all but finitely many n’,
meaning that the set of all n for which Pn fails is a finite set. We leave it as
an easy exercise that ‘Pn is true for all but finitely many n’ is equivalent to
‘Pn is eventually true’.
1. Prove that the inequality
3n + 1
1
√
,
<
100
17n + π
(1)
n2 +
is eventually true. (Any N that works will do. We are not trying to find
the ‘best’ N or the smallest N. We are not trying to find all n for which the
inequality is true.)
Definition: (ε-N, slight variant of Definition 7.1 in Ross) We say that
a real sequence (an )n∈N converges to a real number L, and we write
an → L as n → +∞, or lim an = L, if and only if for every positive
n→+∞
ε (no matter how small) the inequality |an − L| < ε is eventually true.
In other words, an → L as n → +∞ if and only if for every positive
ε (no matter how small) there is an index N such that all terms an
with indices n ≥ N satisfy |an − L| < ε (which means that they are at
distance < ε from the limit L).
If you like quantifiers you can write it like this:
∀ε > 0
∃N ∈ N
no matter how
small ε is
there is a place
in the sequence
such
that
∀n ≥ N
|an − L| < ε
from that
place onwards
all terms are
within ε of L
2. (Ross, Exercise 8.2d) Let (an )n∈N be the sequence defined by the formula
2n + 4
an =
. Prove directly from the ε-N definition of convergence, and
5n + 2
2
without using any Rules of Limits, that an → as n → +∞.
5
1
Definition (M-N). We say that a real sequence (an )n∈N tends to +∞,
and we write an → +∞ as n → +∞, or lim an = +∞, if and only if
n→+∞
for every positive real number M there exists an index N such that,
for all indices n with n ≥ N,
an > M.
(2)
3. Using the M-N definition, and without using any Rules of Limits, prove that
n3 − 100n2 − 200n − 300 → +∞ as n → +∞.
4. Using the M-N definition, and without using any Rules of Limits, prove that
n3 − 3n2 + 5n − 7
→ +∞ as n → +∞.
9n2 − 11n + 13
5. Let (an )n∈N be a sequence of non-negative real
√ numbers converging to a
√
non-negative real number a. Prove that an → a as n → +∞. (Hint)
√
6. (Another proof of the existence of 2) Define a real sequence (an )n∈N as
follows: let a1 be any positive real number and set
2
1
an +
, n = 1, 2, ...
(3)
an+1 =
2
an
Prove the following without using any square roots:
(a) For all integers n with n ≥ 2 we have a2n ≥ 2. (Hint)
(b) For all positive integers n with n ≥ 2 we have an+1 ≤ an . (Hint)
(c) The sequence converges to a positive real number L such that L2 = 2.
2
Hints
Hints for Exercise 5.
Hint 1:
√
√ √
√
√
( x − y)( x + y)
√
√
x− y =
= ···
√
x+ y
Hint 2:
p
√
√
| x − y| ≤ |x − y|, (x, y ≥ 0).
3
Hint for Exercise 6, part (a).
a2n − 2 =
2
1
2
an−1 +
− 2 = · · · = (· · · )2 ≥ 0.
4
an−1
4
Hint for Exercise 6, part (b).
2
1
an +
= ···
an − an+1 = an −
2
an
5