Limit Laws and Monotone Convergence

The $ϵ$ - $M$ definition proves individual limits but is clumsy for computation. The algebraic and order properties below assemble the limit of a complicated expression from the limits of its parts, and the monotone convergence theorem produces a limit from boundedness without naming it in advance. Together they let most limits be computed by inspection.

Limits and inequalities

The first tool compares an unknown sequence to two known ones that close in on a common value.

The squeeze lemma. Two envelopes a_n and b_n close on the common value L; the trapped sequence x_n is forced to the same limit.

Limits respect non-strict inequalities, but they can collapse strict ones.

The strict version is false. With $x_{n} = - 1/ n$ and $y_{n} = 1/ n$ we have $x_{n} < y_{n}$ for every $n$ , yet both limits equal $0$ . Strict inequalities may weaken to non-strict ones under a limit; this is a common source of error.²

The algebra of limits

Limits pass through the four arithmetic operations.

Constant sequences turn these into the familiar corollaries: for $c \in R$ , $lim (c x_{n}) = c lim x_{n}$ and $lim (c + x_{n}) = c + lim x_{n}$ . By induction, $lim x_{n}^{k} = (lim x_{n})^{k}$ for every $k \in N$ . Two further operations commute with limits.

The absolute-value statement is immediate from the reverse triangle inequality $∣ x_{n} ∣ - ∣ x ∣ \leq ∣ x_{n} - x ∣$ : the left side is small whenever the right side is. Its converse fails — $∣ (- 1)^{n} ∣ \to 1$ while ${(- 1)^{n}}$ diverges — so taking absolute values can create convergence that was not there.⁴

The limit laws compose. Given the limits of two input sequences, the limit of any arithmetic combination is the same combination of the limits.

One caution: the laws require the pieces to converge first. The limit $lim_{n \to \infty} (\frac{n ^{2}}{n + 1} - n) = - 1$ is correct only after combining over a common denominator; splitting it as $lim \frac{n ^{2}}{n + 1} - lim n$ is meaningless, since neither piece converges.

A convergence test from the definition

Convergence of ${x_{n}}$ to $x$ is the same as convergence of $∣ x_{n} - x ∣$ to $0$ , which gives a reusable test.

Studying whether ${x_{n}}$ has limit $x$ thus reduces to studying whether the error sequence $∣ x_{n} - x ∣$ goes to zero, which is often a simpler question.

The monotone convergence theorem

The limit laws assume a limit and compute it. Monotone convergence produces a limit from a structural hypothesis, without any candidate value in hand, and it is the first result here to draw on the least-upper-bound property of $R$ .

An increasing sequence is automatically bounded below by its first term, so only an upper bound needs checking; symmetrically for decreasing sequences. The value of the theorem is that it certifies a limit exists even when computing the supremum is hard.

A bounded monotone increasing sequence. Terms climb but never exceed the supremum, and they close on it: past any band below the sup, all terms enter.

A bounded monotone decreasing sequence falls toward its infimum. Past any band above the inf, all terms enter, mirroring the increasing case.

The order of the argument is mandatory: establish convergence first, then solve for the limit. Assuming a limit blindly can produce a wrong answer for a divergent sequence.

Standard limits and the ratio test

A few limits recur so often they are worth settling once. Each follows from the behavior of geometric powers.

The ratio of consecutive terms of ${c^{n}}$ is exactly $c$ . Generalizing to any sequence whose consecutive ratios tend to a limit gives a broadly applicable test.

When $L = 1$ the test gives no information: the four sequences $1/ n \to 0$ , the constant $1 \to 1$ , $(- 1)^{n}$ divergent, and $n$ unbounded all have $L = 1$ yet behave differently. The proof compares ${x_{n}}$ to a geometric sequence ${r^{n}}$ with $L < r < 1$ : past some $M$ the ratios stay below $r$ , so $∣ x_{n} ∣ \leq ∣ x_{M} ∣ r^{- M} r^{n}$ , and the right side goes to zero.⁸

The ratio test for sequences, read off the limit L of consecutive ratios. The boundary case L equal to one carries no information.

Two consequences of the ratio test, used throughout the series lessons:

$n \to \infty lim \frac{2 ^{n}}{n !} = 0$ , since the ratio of consecutive terms is $\frac{2}{n + 1} \to 0 < 1$ .
$n \to \infty lim n^{1/ n} = 1$ . Given $ϵ > 0$ , the ratio test applied to $\frac{n}{( 1 + ϵ ) ^{n}}$ shows that sequence tends to $0$ , so eventually $n < (1 + ϵ)^{n}$ , i.e. $n^{1/ n} < 1 + ϵ$ ; and $n^{1/ n} \geq 1$ always.

Sequence	Behavior	Reason
$c^{n}$ , $0 < c < 1$	$\to 0$	geometric decay
$c^{n}$ , $c > 1$	unbounded	reciprocal of the above
$2^{n} / n!$	$\to 0$	ratio $\frac{2}{n + 1} \to 0$
$n^{1/ n}$	$\to 1$	ratio test on $n / (1 + ϵ)^{n}$
$1/ n^{p}$ , $p > 0$	$\to 0$	squeeze / powers of $1/ n$

Summary

The limit laws let a limit be computed by decomposing an expression into pieces with known limits, provided each piece converges. The squeeze lemma and the null-sequence comparison handle expressions too tangled to decompose. The monotone convergence theorem, which rests on completeness, gives a limit from boundedness alone, and the ratio test settles the geometric and factorial sequences that dominate the study of series.

Lebl, §2.2, Lemma 2.2.1. ↩
Lebl, §2.2, Lemma 2.2.3 and the remark following Corollary 2.2.4. ↩
Lebl, §2.2, Proposition 2.2.5. ↩
Lebl, §2.2, Propositions 2.2.6 and 2.2.7. ↩
Lebl, §2.2, Proposition 2.2.10. ↩
Lebl, §2.1, Theorem 2.1.10. ↩
Lebl, §2.2, Proposition 2.2.11. ↩
Lebl, §2.2, Lemma 2.2.12. ↩