[THEOREMS] Sequences and limits

Algebra and continuity of sequence limits

If $a_n\to K$ and $b_n\to L$ , with $K$ and $L$ finite, then

a_n\pm b_n\to K\pm L,\qquad a_nb_n\to KL,\qquad \frac{a_n}{b_n}\to\frac KL\quad(L\ne0).

If $h$ is continuous at $K$ , then

a_n\to K\quad\Longrightarrow\quad h(a_n)\to h(K).

For example,

\frac{4n^2-3n+2}{2n^2+6n+1} =\frac{4-3/n+2/n^2}{2+6/n+1/n^2} \longrightarrow2.

If a real function has a limit as $x\to\infty$ , its sampled sequence $f(n)$ has the same limit. The converse is false: $f(x)=\sin(\pi x)$ gives $f(n)=0$ for every integer $n$ , although $f(x)$ has no limit as $x\to\infty$ .

[1]:Convergence of a sequence
[2]:Simplify a sequence limit

Squeeze theorem for sequences

If, for all sufficiently large $n$ ,

a_n\le b_n\le c_n,

and $a_n\to L$ and $c_n\to L$ , then $b_n\to L$ .

Only the tail needs to satisfy the inequalities. This helps with growth estimates when an exact formula for the middle sequence is unavailable.

[1]:Convergence of a sequence
[2]:A factorial sequence

Completeness of the real numbers

Every non-empty subset of $\mathbb R$ that is bounded above has a supremum and every non-empty subset bounded below has an infimum.

The characterisation

M=\sup S

is equivalent to:

$x\le M$ for every $x\in S$ ;
for every $\varepsilon>0$ , some $x\in S$ satisfies $M-\varepsilon<x$ .

The second condition says that no smaller number remains an upper bound. Reversing the inequalities gives the corresponding characterisation of an infimum.

[1]:Supremum and infimum

Monotone convergence theorem

A non-decreasing sequence bounded above converges. A non-increasing sequence bounded below converges.

The theorem guarantees a finite limit without giving a formula for it. Completeness supplies the least upper bound or greatest lower bound that becomes the limit.

Theorems from Sequences and limits

Algebra and continuity of sequence limits

Squeeze theorem for sequences

Completeness of the real numbers

Monotone convergence theorem

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