3. Sequences

Def: Functions of the form $a:\mathbb{N}\to \mathbb{R}$

Convergence (epsilon criterium): $\underset{n\to \infty }{\lim }{x}_n=L\in \mathbb{R}⟺\forall \epsilon >0\exists N\in \mathbb{N}\forall n\ge N:|x_n-L|<\epsilon$

Subsequence: sequence obtained by selecting infinitely many $a_n$ with increasing $n$ Accumulation point: limit of a subsequence Limit superior/inferior: largest/smallest accumulation point ($\underset{n\to \infty }{\lim }\sup \{a_k\mid k\ge n\}$ and v.v.)

Bounded: set of all series elements has upper and lower bound Monotonically increasing/decreasing: $\forall n,m\in {\mathbb{N}}_0:m>n\Rightarrow a_m\le a_n$

Harmonic: $\frac{1}{a_{n+1}}-\frac{1}{a_n}=d$ for some constant $d$ Geometric: $a_n=d\cdot q^n$ for some constants $d,q$

Monotone Convergence Theorem

Theorem

If a sequence $(a_n)$

• is monotone increasing and bounded above, or
• is monotone decreasing and bounded below

then the sequence converges.

Sandwich Theorem

Intuitively: If a sequence is trapped between two sequences converging to the same limit, it must converge to that limit too.

Cauchy Sequence

$$\forall \epsilon >0,\exists N\text{ s.t. }m,n\ge N⟹|S_n-S_m|<\epsilon$$

Lemma

A sequence converges $⟺$ it is a Cauchy sequence