2Convergence of sequences
IA Analysis I
2.1 Definitions
Definition (Sequence). A sequence is, formally, a function
a
:
N → R
(or
C
).
Usually (i.e. always), we write
a
n
instead of
a
(
n
). Instead of
a
, we usually write
it as (a
n
), (a
n
)
∞
1
or (a
n
)
∞
n=1
to indicate it is a sequence.
Definition (Convergence of sequence). Let (
a
n
) be a sequence and
` ∈ R
. Then
a
n
converges to
`
, tends to
`
, or
a
n
→ `
, if for all
ε >
0, there is some
N ∈ N
such that whenever n > N , we have |a
n
− `| < ε. In symbols, this says
(∀ε > 0)(∃N)(∀n ≥ N) |a
n
− `| < ε.
We say ` is the limit of (a
n
).
One can think of (∃N)(∀n ≥ N) as saying “eventually always”, or as “from
some point on”. So the definition means, if
a
n
→ `
, then given any
ε
, there
eventually, everything in the sequence is within ε of `.
We’ll now provide an alternative form of the Archimedean property. This is
the form that is actually useful.
Lemma (Archimedean property v2). 1/n → 0.
Proof.
Let
ε >
0. We want to find an
N
such that
|
1
/N −
0
|
= 1
/N < ε
. So pick
N
such that
N >
1
/ε
. There exists such an
N
by the Archimedean property v1.
Then for all n ≥ N , we have 0 < 1/n ≤ 1/N < ε. So |1/n − 0| < ε.
Note that the red parts correspond to the definition of convergence of a
sequence. This is generally how we prove convergence from first principles.
Definition (Bounded sequence). A sequence (a
n
) is bounded if
(∃C)(∀n) |a
n
| ≤ C.
A sequence is eventually bounded if
(∃C)(∃N)(∀n ≥ N) |a
n
| ≤ C.
The definition of an eventually bounded sequence seems a bit daft. Clearly
every eventually bounded sequence is bounded! Indeed it is:
Lemma. Every eventually bounded sequence is bounded.
Proof.
Let
C
and
N
be such that (
∀n ≥ N
)
|a
n
| ≤ C
. Then
∀n ∈ N
,
|a
n
| ≤
max{|a
1
|, ··· , |a
N−1
|, C}.
The proof is rather trivial. However, most of the time it is simpler to prove
that a sequence is eventually bounded, and this lemma saves us from writing
that long line every time.