Foundations — Squeeze theorem (sandwich theorem)

This page assumes you know nothing. Before we can even read the parent note's first line, we must earn every symbol it throws at us. We go one brick at a time, and each brick sits on the one before it.

1. What is a function? The symbol $f(x)$

The letter $x$ is just a name for whatever number we put in. The letter $f$ is the name of the machine. So $f(x)$ is read aloud as "f of x" — NOT "f times x". There is no multiplication here at all.

Think of a vending machine: press button $x=3$, out drops item $f(3)$. Different button, different snack.

Why the topic needs this: the squeeze theorem talks about three machines $f$, $g$, $h$ at once. If $f(x)$ is a mystery, everything after is a mystery.

2. The symbol $\le$ and what "trapped between" looks like

Picture the number line, a straight ruler stretching left (small) to right (big):

• $a<b$: $a$ sits somewhere to the left of $b$.
• $g(x)\le f(x)\le h(x)$ reads as a single sentence: at every input $x$, the output $g(x)$ is on the left, $f(x)$ is in the middle, $h(x)$ is on the right. The middle output is sandwiched.

Why the topic needs this: the entire hypothesis of the theorem is one chain of $\le$ signs. Misreading it breaks everything.

3. The idea of a limit: the symbol ${\lim }_{x\to a}f(x)=L$

Let us decode each piece of the new symbol, because it is dense:

• $x\to a$ : the little arrow means "approaches". $x$ is a wanderer sneaking up on the fixed target $a$, from the left side and the right side, but never actually landing on $a$.
• $L$ : the letter we reserve for the value the outputs settle down to. (Just a name, like calling the destination "the 3rd floor".)
• $\lim$ : short for limit, the question "where is $f$ heading?"

Why the topic needs this: the theorem's conclusion is a limit statement. Its hypotheses are two limit statements. No limit idea, no theorem.

4. The precise version: $\epsilon$ and $\delta$

The proof in the parent note uses the Limit definition (epsilon-delta). Two Greek letters appear; here is what they are before we ever use them.

The heart of it: no matter how tiny a target band $\epsilon$ you draw around $L$, I can find an input band $\delta$ around $a$ so that every $x$ in my band lands its output inside your band.

• $|x-a|$ means "the distance between $x$ and $a$" (the two vertical bars = absolute value = distance, always $\ge 0$, drops the minus sign).
• $0<|x-a|<\delta$ means "$x$ is within distance $\delta$ of $a$, but not equal to $a$" (the leftmost $0<$ forbids landing on $a$).
• $|f(x)-L|<\epsilon$ means "the output is within distance $\epsilon$ of $L$".

Why the topic needs this: the proof (Steps 1–5 in the parent) is written entirely in $\epsilon$, $\delta$, $|\cdot |$ and $\min$. These are its only tools.

5. Bounded, oscillating, and "vanishing"

Three plain-English notions the examples lean on.

Why the topic needs this: the squeeze theorem is at its most powerful exactly when a function both oscillates (so you can't plug in) and is trapped by a vanishing wall.

6. Sequences and $n\to \infty$

The squeeze theorem works here too — the sandwich picture is identical, we just swap "$x$ creeps toward $a$" for "$n$ marches to infinity".

Why the topic needs this: Example 3 in the parent uses this version to kill $\cos n/n$.

Prerequisite map

Worked micro-check: reading a full statement

Equipment checklist

Self-test: can you answer each before revealing? If any stumps you, reread its section.

Connections

• Squeeze theorem (sandwich theorem) — the parent this page equips you for.
• Limit definition (epsilon-delta) — the $\epsilon$–$\delta$ machinery decoded in §4.
• Limits of trigonometric functions — where bounded sine meets the squeeze.
• Bounded times vanishing — the pattern behind §5.
• Oscillating functions — why we can't just plug in.
• Limits of sequences — the $n\to \infty$ version of §6.
• Continuity — the next idea once limits are secure.