Foundations — Sequences — convergence, divergence, boundedness, monotonicity

Before you can read the parent note, you need to own every symbol it throws at you. Below, each symbol gets three things: plain words, the picture, and why the topic needs it. They are ordered so each one leans only on the ones above it.

1. The counting numbers $\mathbb{N}$

The picture: a row of evenly spaced ticks marching rightward off the edge of the page — position 1, position 2, position 3… These are the slots a sequence fills. They are the "$n$" that will "march off to infinity" (a phrase we make precise in Section 2b).

Why the topic needs it: a sequence is a list, and a list needs numbered slots. $\mathbb{N}$ supplies the slot numbers.

2. Ordering on $\mathbb{N}$ and $\mathbb{R}$: $<$, $>$, $\le$, $\ge$

The picture: point at two dots on the number line; the one further left is the smaller. "$\epsilon >0$" simply means "$\epsilon$ sits strictly to the right of $0$" — i.e. it is positive.

Why the topic needs it first: every later statement — "$\epsilon >0$", "$n>N$", "$a_n\le M$" — is a comparison. We must fix what "greater/less" means before using it. Note $<,>$ are strict (equality forbidden); $\le ,\ge$ allow equality.

2b. What "$\to \infty$" means — and the symbol $\infty$

The picture: the ticks of $\mathbb{N}$ streaming off the right edge of the page forever. There is no last tick; $\infty$ names that "keeps going" behaviour, marked as an open horizon, never a dot you land on.

Why the topic needs it: the one question of the whole subject is "what happens as $n$ grows without bound?" The symbol $\infty$ lets us write that compactly (in $(a_n{)}_{n=1}^{\infty }$ and ${\lim }_{n\to \infty }$), but you must always mentally translate it back to "for slots as far right as you like".

3. The real numbers $\mathbb{R}$ and the number line

The picture: a horizontal line with $0$ in the middle, positive numbers ($>0$) to the right, negative ($<0$) to the left. Unlike $\mathbb{N}$'s isolated ticks, $\mathbb{R}$ has no holes — between any two points there is always another.

Why the topic needs it: the values a sequence takes live here. Slot numbers come from $\mathbb{N}$ (top ruler in the figure); the values they point to are real numbers (bottom line). The "gap-free" property is not decorative — it is the deep fact (Completeness of the Real Numbers) that later powers the Monotone Convergence Theorem.

4. A function $a:\mathbb{N}\to \mathbb{R}$

The picture: every tick $n$ on the top ruler shoots an arrow down to exactly one point on the number line. Input $1\to a_1$, input $2\to a_2$, and so on. A sequence is literally this bundle of arrows.

Why the topic needs it: calling a sequence a "function" is not pedantry — it tells us the rules of functions apply (each input gets exactly one output; we can ask about long-run behaviour just like Limits of functions but only at integer inputs).

5. The term $a_n$ and the whole sequence $(a_n{)}_{n=1}^{\infty }$

The picture: $a_n$ is one dot; $(a_n)$ is the whole scatter of dots. The parentheses are the "box" that says treat all of them together.

Why the topic needs it: convergence is a property of the whole list, never of one term. We need one symbol for a single value ($a_n$) and a different symbol for the list-as-a-thing ($(a_n)$).

6. The subscript index $n$ vs. a fixed cutoff $N$, and the tail

The picture: imagine a vertical dashed fence dropped at slot $N$. Everything to the right of the fence is "the tail". $n$ is a walker; $N$ is the fence post.

Why the topic needs it: the whole definition of convergence is a statement about the tail — "past some fence, every term behaves". The first $N$ terms are allowed to misbehave; only $(a_n{)}_{n>N}$ matters. Confusing moving-$n$ with frozen-$N$ makes the definition unreadable.

7. Absolute value $|x|$ — distance

The picture: the length of the gap between two dots, always positive, no matter which is on the left.

Why the topic needs it: to say "$a_n$ gets close to a target value" we first need a name for that target and a way to measure closeness. Absolute value is the tool that turns "close" into a measurable, sign-free number.

8. The target value $L$

The picture: one special dot on the number line, and we watch whether the sequence's dots pile up on top of it. In figure s03 it is the solid central horizontal line the terms squeeze toward.

Why the topic needs it: "do the terms settle on a single number?" only makes sense once we name that single number. $L$ is that name. The distance $|a_n-L|$ from Section 7 measures how far term $n$ still is from the target.

9. The tolerance $\epsilon$ and the band around $L$

The picture: two horizontal guide-lines a height $\epsilon$ above and below $L$, forming a horizontal stripe. The statement $|a_n-L|<\epsilon$ means the dot $a_n$ sits inside the stripe.

Why the topic needs it: "close" is vague; a stripe of any width you name is precise. If we can trap the whole tail in every stripe — even the razor-thin ones — the terms genuinely home in on $L$.

10. The quantifiers $\forall$ and $\exists$

The picture: $\forall$ is the challenger trying every stripe width; $\exists$ is you finding one fence that works for the width in hand. The order matters: the challenger moves first ($\epsilon$), then you respond ($N$) — so your $N$ may depend on their $\epsilon$.

Why the topic needs it: the convergence definition is exactly a two-move game written with these symbols: $\forall \epsilon >0,\exists N:n>N⟹|a_n-L|<\epsilon .$ Read it aloud: "for any stripe, there is a fence past which every tail term lies inside." Every symbol in that line is now defined.

11. Implication $⟹$, the limit symbol $\lim$, and divergence

The picture: the arrow $⟹$ is a one-way street — being past the fence forces being inside the stripe (not the reverse). $\lim =L$ is the single word "converges" wearing a formula costume.

Why the topic needs it: "converges" and "diverges" are the two verdicts the whole subject delivers. You cannot answer the core question without a precise name — and criterion — for the "no, they never settle" case.

12. Bounds: $M$, $m$, $K$, $\sup$, $\inf$

The picture: two horizontal rails, a ceiling and a floor, with all the dots sandwiched between. $\sup$ is the ceiling pushed down until it just touches the dots; $\inf$ is the floor pushed up until it just touches.

Why the topic needs it: boundedness is a necessary condition for convergence (a sequence diverging to $\infty$ can't settle). And $\sup$/$\inf$ are the exact values a monotone bounded sequence lands on — the heart of the Monotone Convergence Theorem, guaranteed to exist by Completeness of the Real Numbers.

13. Monotone comparisons: $a_{n+1}$ vs $a_n$

The picture: a staircase that only ever goes up (or only ever goes down) — never a wobble. Contrast $(-1{)}^n$: pure zig-zag, not monotone, and (from Section 11) divergent.

Why the topic needs it: monotone + bounded is the winning combination that forces convergence. To test monotonicity you look at the sign of $a_{n+1}-a_n$, so you must first know exactly what $a_{n+1}$ means.

How these feed the topic

Read top to bottom: the number systems and ordering build the notion of a term and the "$\to \infty$" horizon; the term plus a moving index plus distance plus a target $L$ plus quantifiers build the convergence definition (and its opposite, divergence); bounds plus monotonicity plus that definition build the payoff theorem.

Equipment checklist

Test yourself — cover the right side of each ::: line and answer out loud.