So far, we’ve been considering the case where a sequence converges to a real number *L*. It’s also possible for a sequence to approach +∞ or -∞. The infinity symbol “∞” should be thought of as a convenient symbol instead of an actual infinity; in particular, it bears no relation to the cardinality (i.e. size) of the set of integers.

Definition. Let (a_{n}) be a sequence of real numbers. We write (a_{n}) → ∞ if

- for every real M, there exists N such that if n>N, then a
_{n}>M.By the same token, (a

_{n}) → -∞ if

- for every real M, there exists N such that if n>N, then a
_{n}<M.

[ A sequence which approaches +∞. ]

Let’s define a shorthand here: we say that “** X eventually happens**” for a sequence (

*a*) if there exists

_{n}*N*such that for all

*n*>

*N*,

*X*happens for

*a*. It should be apparent that if “

_{n}*X*eventually happens” and “

*Y*eventually happens” are both true, then “(

*X*and

*Y*) eventually happen” is also true.

Warning: if we have *infinitely* many statements of the form “for every *m*,* **X _{m}* eventually happens”, one should not write “

*X*eventually happens for all

_{m}*m*” due to ambiguity in the English language. The latter statement can also be interpreted as “eventually, we have (

*X*and

_{1}*X*and

_{2}*X*…)”.

_{3}To illustrate the difference, consider the definition of (*a _{n}*) → ∞ which says: for every

*m*, eventually

*a*>

_{n}*m*. This is different from saying: we eventually have (

*a*>

_{n}*m*for all

*m*) since under this interpretation, no sequence of real numbers can satisfy it.

Now the three different types of limits can be summarised as follows:

We write (*a _{n}*) → if for every we eventually have

**Properties**.

The limits satisfy the following properties, as expected from intuition.

- If (
*a*) → ∞ or_{n}*L*, and (*b*) → ∞, then (_{n}*a*+_{n}*b*) → ∞._{n} - If (
*a*) → ∞ or_{n}*L*>0, and (*b*) → ∞, then (_{n}*a*_{n}*b*) → ∞._{n} - If (
*a*) → -∞ or_{n}*L*<0, and (*b*) → ∞, then (_{n}*a*_{n}*b*) → -∞._{n} - If (
*a*) → ∞ or -∞, then (1/_{n}*a*) → 0._{n} - If (
*a*) → 0 for a sequence of_{n}*positive*numbers (*a*), then (1/_{n}*a*) → ∞._{n}

**Proof (Sketch)**.

To prove the first property, suppose (*a _{n}*) →

*L*;

- we eventually have |
*a*–_{n}*L*| < 1, so*a*>_{n}*L*-1; - for any
*M*, we eventually have*b*>_{n}*M*-(*L*-1).

Hence, for any *M* we eventually have *a _{n}*+

*b*>

_{n }*M*which proves that (

*a*+

_{n}*b*) → ∞. For the case (

_{n}*a*) → ∞, replace the statement at the first bullet with “eventually,

_{n}*a*>0″.

_{n}For the second property: suppose (*a _{n}*) →

*L*>0. Eventually, |

*a*–

_{n}*L*| <

*L*/2, so

*a*>

_{n}*L*/2, and for any

*M*>0, eventually

*b*>2

_{n}*M*/

*L*. Hence, for any

*M*, eventually

*a*

_{n}*b*>

_{n}*M*. This proves that (

*a*

_{n}*b*) → ∞.

_{n}The third property is similar to the second. Since (*a _{n}*) →

*L*<0, eventually we have |

*a*–

_{n}*L*| < –

*L*/2, so

*a*<

_{n}*L*/2. For any

*M*<0, we also eventually have

*b*>2

_{n}*M*/

*L*. Thus

*a*

_{n}*b*<

_{n}*M*eventually.

For the fourth property: suppose (*a _{n}*) → ∞ or -∞. In the first case, for any

*M*>0, we eventually have

*a*>

_{n}*M*>0, which gives 0 < 1/

*a*<1/

_{n }*M*. Thus, for any ε>0, we eventually have 0 < 1/

*a*< ε. This proves that (

_{n }*a*) → 0.

_{n}We’ll leave the remaining cases to the reader. ♦

**Corollaries**

- If (
*a*) → ∞ (resp. -∞), then (-_{n}*a*) → -∞ (resp. ∞). [ Multiply_{n}*a*by the constant sequence of (-1). ]_{n} - If (
*a*) → ∞ or_{n}*L*, and (*b*) → -∞, then (_{n}*a*–_{n}*b*) → ∞. [ Write_{n}*a*–_{n}*b*=_{n }*a*+(-_{n}*b*) and apply two properties. ]_{n}

**More Properties**.

Correspondingly, we now have:

- (Monotone convergence) An
*unbounded*increasing sequence (*a*) must approach ∞. [In conjunction with the earlier result, an increasing sequence must have a limit_{n}*L*or ∞.] - (Convergence implies bounded) If (
*a*) → ∞, then (_{n}*a*) has a lower bound._{n} - (Squeeze theorem) If (
*a*) and (_{n}*b*) are two sequences such that_{n}*a*≥_{n}*b*and (_{n}*b*) → ∞, then (_{n}*a*) → ∞._{n}

**Proof**.

- Pick
*M*. Since (*a*) is unbounded, there exists_{n}*N*such that*a*>_{N}*M*. Hence, eventually we have*a*≥_{n}*a*>_{N}*M*. So (*a*) → ∞ as expected._{n} - Eventually we have
*a*>0. Since only finitely many terms are left out, (_{n}*a*) has a lower bound._{n} - For any
*M*, eventually we have*b*≥_{n}*M*. This implies*a*≥_{n}*b*≥_{n}*M*. ♦

We’ll leave it to the reader to write down the corresponding cases for (*a _{n}*) → -∞.

## Limits Inferior and Superior

Let (*a _{n}*) be a sequence of real numbers.

Definition. Define a new sequence . Clearly since we’re taking the sup of fewer elements as we go along. Since (b_{n}) is a decreasing sequence, it must have a limit. Thelimit superiorof (a_{n}), written , is defined by .By the same token, let . Now so it has a limit, which is the

limit inferiorof (a_{n}), written .

Here’s a pictorial representation of lim sup and lim inf.

Let’s look at lim sup and lim inf in further detail.

First suppose (*a _{n}*) isn’t upper-bounded. Now even if we drop finitely many terms, the resulting sequence still isn’t upper-bounded. This gives . In this case, the limit superior is +∞.

If (*a _{n}*) is upper-bounded, then each

*b*is a real number and we get a decreasing sequence (

_{n}*b*). The limit of this sequence may be finite or -∞. If it’s -∞, for any

_{n}*M*, eventually

*b*<

_{n}*M*and hence

*a*≤

_{n}*b*<

_{n}*M*. So the lim sup = -∞ only occurs if

*a*→ -∞.

_{n }*In particular, observe that the lim sup is well-defined for all sequences.*

In summary, we have the following cases:

**Case 1 – (**: lim sup (*a*) is not upper-bounded_{n}*a*) = +∞._{n}**Case 2 – (**: lim sup (*a*) → -∞_{n}*a*) = -∞._{n}**Case 3 – otherwise**: lim sup (*a*) =_{n}*L*, a real value.

[ *Warning: even if ( a_{n}) is not lower-bounded, it’s possible for its lim sup to be finite. See the third example later*. ]

By the same token, for the **limit inferior** of (*a _{n}*) we get three cases.

**Case 1 –**each (**(**:*a*) is not lower-bounded_{n}*c*) = -∞, so lim inf (_{n}*a*) = -∞._{n}**Case 2 – (**: lim inf (*a*) → +∞_{n}*a*) = +∞._{n}**Case 3 – otherwise**: lim inf (*a*) =_{n}*L*, a real value.

**Examples**

- Suppose is an
*increasing*sequence which converges to a finite*L*.Suppose*a*= (-1)_{n }^{n}, i.e. the sequence is -1, +1, -1, +1, … . Then we have: for all*n*,*b*= +1 and_{n}*c*= -1. Thus, lim sup (_{n}*a*) = +1, lim inf (_{n}*a*) = -1._{n}- Each subsequence is an increasing sequence converging to
*L*. - From the proof of monotone convergence theorem, sup(
*a*) = lim(_{n}*a*) =_{n}*L*. Thus and lim sup (*a*) =_{n}*L*. - On the other hand,
*c*=_{n}*a*. Thus lim inf (_{n}*a*) = lim (_{n}*c*) =_{n}*L*.

- Each subsequence is an increasing sequence converging to
- Take the sequence
*a*= (0, -1, 0, -2, 0, -3, 0, -4, …). Since the sequence is not lower-bounded, we have lim inf (_{n }*a*) = -∞. On the other hand, since each_{n}*b*= 0 we have lim sup (_{n}*a*) = 0._{n}

**Basic Properties**.

Pick any sequences (*a _{n}*) and (

*b*). Then:

_{n}- if
*c*> 0; - if
*c*< 0; - . [
*If you can’t recall which way the inequality goes, just think of the two alternating sequences (1, 0, 1, 0, 1, 0, …) and (0, 1, 0, 1, 0, 1, …). Then LHS = 1 and RHS = 1+1 = 2.*]

**Proof (a bit sketchy)**.

- This follows from if
*c*>0. - Follows from if
*c*<0. - Let and . Then for each
*m*≥*n*, , and thus . Hence, . Take the limit of both sides. ♦

**Theorem**.

Let (*a _{n}*) be a sequence.

- If lim (
*a*) =_{n}*L*(resp. +∞, -∞), then lim inf (*a*) = lim sup (_{n}*a*) =_{n}*L*(resp. +∞, -∞). - Conversely, if lim inf (
*a*) = lim sup (_{n}*a*) =_{n}*L*(resp. +∞, -∞), then lim (*a*) =_{n}*L*(resp. +∞, -∞).

**Proof**

For the first statement, consider the case of finite limit. Given ε>0, for some *N*, we have *L*-(ε/2) < *a _{n}* <

*L*+(ε/2) for all

*n>*

*N*. So

*L*+(ε/2) is an upper bound for

*a*,

_{n}*a*,

_{n+1}*a*…. and

_{n+2}*b*, being the least upper bound of these values, satisfy

_{n}*b*≤

_{n}*L*+(ε/2) <

*L*+ε. Next, since

*a*>

_{n}*L*-(ε/2), we have

*b*≥

_{n}*a*>

_{n}*L*-ε. This gives |

*b*–

_{n}*L*| < ε for all

*n*>

*N*as desired. So lim sup (

*a*) =

_{n}*L*. The case where lim inf (

*a*) =

_{n}*L*is similar, or we can just replace (

*a*) by (-

_{n}*a*).

_{n}

For the second statement, let ε>0. Eventually we have:

In particular, and . Hence, eventually which proves that lim (*a _{n}*) =

*L*. ♦