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→Rf: \N \to \R. We use the notation (an)n∈N(a_n)_{n\in\N} for a real-valued sequence and f(n)=anf(n) = a_n for the values of ff at nn.

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 (an)n∈N(a_n)_{n\in\N} be a sequence. (an)n∈N(a_n)_{n\in\N} is said to be convergent if there exists a number a∈Ra \in \R s.t. for any ε\epsilon there exists N∈NN \in \N s.t. ∣an−a∣<ε\abs{a_n - a} < \epsilon for any n≥Nn \geq \N.
Note: The number aa is called the limit of (an)n∈N(a_n)_{n\in\N}. We write lim⁡n→∞=a\lim_{n\to\infty} = a or an→n→∞aa_n \xrightarrow{n\to\infty} a.
Example (Convergent sequence): Let an=1na_n = \frac{1}{n}. We show that (an)n∈N(a_n)_{n\in\N} is convergent with limit 00. Let ε>0\epsilon > 0 be an arbitrary strictly positive real number. As we know from chapter 1 that for any x∈Rx \in \R there exists n∈Nn \in \N s.t. n>xn > x, we pick N∈NN \in \N, we pick N∈NN \in \N, s.t. N>1εN > \frac{1}{\epsilon}. Then, for any n≥Nn \geq N, ∣an−0∣=1n≤1N<ε\abs{a_n - 0} = \frac{1}{n} \leq \frac{1}{N} < \epsilon.
Proposition 3.3 (Uniqueness of limit): If (an)n∈N(a_n)_{n\in\N} is convergent, then its limit aa is unique.

3.1.2 Results on Real-valued Sequences

Definition 3.4 (Bounded sequence): A sequence (an)n∈N(a_n)_{n\in\N} is said to be bounded if there exists a real number M>0M > 0 s.t. for any n∈Nn\in\N, ∣an∣≤M\abs{a_n} \leq M. (an)n∈N(a_n)_{n\in\N} is said to be bounded from below, resp. above, if there exists M∈RM \in \R s.t. an≥Ma_n \geq M, resp. an≤Ma_n \leq M, for any n∈Nn \in \N.
Proposition 3.5 (Convergent implies bounded): If (an)n∈N(a_n)_{n\in\N} is convergent, then it is bounded.
Definition 3.6 (Increasing, decreasing sequence): A sequence (an)n∈N(a_n)_{n\in\N} is increasing, resp. decreasing, if an≤an+1a_n \leq a_{n+1}, resp. an≥an+1a_n \geq a_{n+1}, ∀n∈N\forall n \in \N. (an)n∈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 (an)n∈N(a_n)_{n\in\N} is increasing, resp. decreasing, with limit aa, we write an↑aa_n \uparrow a, resp. an↓aa_n \downarrow a.
Proposition 3.8 (Bounded, monotonic implies convergent): A bounded and monotonic sequence (an)n∈N(a_n)_{n\in\N} is convergent.
Example (Bounded, decreasing implies convergent): Let ∣r∣<1\abs{r} < 1, and consider an=∣r∣na_n = \abs{r}^n, n∈Nn \in \N. For any n∈Nn \in \N, an<1a_n < 1 and ∣r∣n+1=∣r∣n∣r∣≤∣r∣n\abs{r}^{n+1} = \abs{r}^n \abs{r} \leq \abs{r}^n. Thus, (an)n∈N(a_n)_{n\in\N} is bounded and decreasing and hence there exists an LL s.t. lim⁡n→∞an=L\lim_{n\to\infty} a_n = L.
Proposition 3.9 (Arithmetic of limits): Let (an)n∈N(a_n)_{n\in\N} and (bn)n∈N(b_n)_{n\in\N} be two convergent sequences s.t. an→n→∞aa_n \xrightarrow{n\to\infty} a and bn→n→∞bb_n\xrightarrow{n\to\infty} b. The following properties hold:
  1. an+bn→n→∞a+ba_n + b_n \xrightarrow{n\to\infty} a+b
  2. anbn→n→∞aba_n b_n \xrightarrow{n\to\infty} ab
  3. anbn→n→∞ab\frac{a_n}{b_n} \xrightarrow{n\to\infty} \frac{a}{b} if b≠0b\neq 0

TODO