Intro to Probability

3. Sequences

3.1 Real-valued Sequences

3.1.1 Introductory Terms

Definition 3.1 (Real-valued sequence): A real-valued sequence is a function $f: \N \to \R$. We use the notation $(a_n)_{n\in\N}$ for a real-valued sequence and $f(n) = a_n$ for the values of $f$ at $n$.

In this section we treat only real-valued sequences. Thus, for now, a sequence is a real-valued sequence.

Definition 3.2 (Convergent sequence): Let $(a_n)_{n\in\N}$ be a sequence. $(a_n)_{n\in\N}$ is said to be convergent if there exists a number $a \in \R$ s.t. for any $\epsilon$ there exists $N \in \N$ s.t. $\abs{a_n - a} < \epsilon$ for any $n \geq \N$.
Note: The number $a$ is called the limit of $(a_n)_{n\in\N}$. We write $\lim_{n\to\infty} = a$ or $a_n \xrightarrow{n\to\infty} a$.
Example (Convergent sequence): Let $a_n = \frac{1}{n}$. We show that $(a_n)_{n\in\N}$ is convergent with limit $0$. Let $\epsilon > 0$ be an arbitrary strictly positive real number. As we know from chapter 1 that for any $x \in \R$ there exists $n \in \N$ s.t. $n > x$, we pick $N \in \N$, we pick $N \in \N$, s.t. $N > \frac{1}{\epsilon}$. Then, for any $n \geq N$, $\abs{a_n - 0} = \frac{1}{n} \leq \frac{1}{N} < \epsilon$.
Proposition 3.3 (Uniqueness of limit): If $(a_n)_{n\in\N}$ is convergent, then its limit $a$ is unique.

3.1.2 Results on Real-valued Sequences

Definition 3.4 (Bounded sequence): A sequence $(a_n)_{n\in\N}$ is said to be bounded if there exists a real number $M > 0$ s.t. for any $n\in\N$, $\abs{a_n} \leq M$. $(a_n)_{n\in\N}$ is said to be bounded from below, resp. above, if there exists $M \in \R$ s.t. $a_n \geq M$, resp. $a_n \leq M$, for any $n \in \N$.
Proposition 3.5 (Convergent implies bounded): If $(a_n)_{n\in\N}$ is convergent, then it is bounded.
Definition 3.6 (Increasing, decreasing sequence): A sequence $(a_n)_{n\in\N}$ is increasing, resp. decreasing, if $a_n \leq a_{n+1}$, resp. $a_n \geq a_{n+1}$, $\forall n \in \N$. $(a_n)_{n\in\N}$ is said to be monotonic if it is either increasing or decreasing.
Definition 3.7 (Increasing, decreasing limit): If $(a_n)_{n\in\N}$ is increasing, resp. decreasing, with limit $a$, we write $a_n \uparrow a$, resp. $a_n \downarrow a$.
Proposition 3.8 (Bounded, monotonic implies convergent): A bounded and monotonic sequence $(a_n)_{n\in\N}$ is convergent.
Example (Bounded, decreasing implies convergent): Let $\abs{r} < 1$, and consider $a_n = \abs{r}^n$, $n \in \N$. For any $n \in \N$, $a_n < 1$ and $\abs{r}^{n+1} = \abs{r}^n \abs{r} \leq \abs{r}^n$. Thus, $(a_n)_{n\in\N}$ is bounded and decreasing and hence there exists an $L$ s.t. $\lim_{n\to\infty} a_n = L$.
Proposition 3.9 (Arithmetic of limits): Let $(a_n)_{n\in\N}$ and $(b_n)_{n\in\N}$ be two convergent sequences s.t. $a_n \xrightarrow{n\to\infty} a$ and $b_n\xrightarrow{n\to\infty} b$. The following properties hold:
  1. $a_n + b_n \xrightarrow{n\to\infty} a+b$
  2. $a_n b_n \xrightarrow{n\to\infty} ab$
  3. $\frac{a_n}{b_n} \xrightarrow{n\to\infty} \frac{a}{b}$ if $b\neq 0$

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