Start adding sequences in metric spaces

Author: jhobbs

content/analysis/real-analysis/sequences.md

@@ -8,7 +8,15 @@ title: Sequences
 
 # Sequences
 
-A **sequence** of numbers if a function $f$ from $\mathbb{N}$ to $\mathbb{R}$. Typically, we say $a_1 = f(1), a_2 = f(2), a_3 = f(3), \dots$ and write the sequence as a list $a_1, a_2, a_3, \dots.$ When it's possible to give a rule $f(n)$ for the $n$th term of a sequence, we can write $a_n = f(n)$ and refer to the sequence using just the rule. For example, $a_n = \frac{3n}{n+1} = 3/2, 6/3, 9/4, \dots$
+:::definition "Real Sequence"
+A **real sequence** of numbers if a function $f$ from $\mathbb{N}$ to $\mathbb{R}$.
+:::
+
+:::note
+A @{real sequence|real-sequence} is a special case of a @sequence.
+
+Typically, we say $a_1 = f(1), a_2 = f(2), a_3 = f(3), \dots$ and write the sequence as a list $a_1, a_2, a_3, \dots.$ When it's possible to give a rule $f(n)$ for the $n$th term of a sequence, we can write $a_n = f(n)$ and refer to the sequence using just the rule. For example, $a_n = \frac{3n}{n+1} = 3/2, 6/3, 9/4, \dots$
+:::
 
 We say that a sequence has a limit $L$ if we can make terms arbitrarily close to $L$ by taking $n$ to be sufficiently large. More precisely, the **limit** of a sequence $a_n$ as $n$ approaches infinity is $L$ ($\lim_{n \to \infty} a_n = L$) if for every $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $\|a_n -L\| < \epsilon$ whenever $n \geq N$.
 
@@ -143,4 +151,4 @@ If a sequence $a_n$ is not bounded above, then $\limsup_{n \to \infty}{a_n} = \i
 
 If a sequence $a_n$ is bounded below then $\liminf_{n \to \infty}{a_n} = \lim_{n \to \infty}\inf{\{a_n, a_{n+1}, \dots\}}.$
 
-If a sequence $a_n$ is not bounded below, then $\liminf_{n \to \infty}{a_n} = -\infty.$
+If a sequence $a_n$ is not bounded below, then $\liminf_{n \to \infty}{a_n} = -\infty.$

content/analysis/sequences-and-series.md

@@ -0,0 +1,37 @@
+---
+description: Sequences and Series in Euclidean and Metric Spaces
+layout: page
+title: Sequences and Series
+---
+
+# Sequences
+
+Review the definition of a sequence
+
+@embed{sequence}
+
+:::definition "Converge"
+A sequence $\{p_n\}$ in a @metric-space $X$ is said to **converge** if there is a point $p \in X$ with the following property: For every $\epsilon > 0,$ there is an integer $N$ such that $n \geq N$ implies that $d(p_n, p) < \epsilon.$ 
+:::
+
+:::definition "Limit"
+If a @sequence $\{p_n\}$ converges to $p,$ we say that $p$ is the **limit** of $\{p_n\},$ denoted as:
+
+$$ \lim_{n \to \infty} p_n = p. $$
+:::
+
+:::definition "Diverge"
+If a @sequence $\{p_n\}$ does not @converge, it is said to **diverge.**
+:::
+
+:::definition "Range (sequence)"
+The set of all points $p_n$ of a sequence $\{p_n\} (n = 1, 2, 3, \dots)$ is the **@range** of $\{p_n\}.$
+:::
+
+:::note
+The @{range|range-sequence} of a @sequence may be @finite or it may be @infinite.
+:::
+
+:::definition "Bounded (sequence)"
+The @sequence $\{p_n\}$ is said to be **bounded** if its @{range|range-sequence} is @bounded.
+:::