Style Guide
Every book on this site uses the same small set of boxes for the same purposes, every time. Here’s what each one means when you see it.
Definition
Introduces one or more terms — shown by name, in capitals, so it reads like a dictionary entry. A term highlighted in blue elsewhere on the page always refers back to one of these.
A natural number greater than 1 with no positive divisors other than 1 and itself.
A definition can introduce more than one term at once:
A function $f: A \to B$ is injective if distinct inputs always map to distinct outputs, surjective if every element of $B$ is hit by some input, and bijective if it is both.
Theorem and Proof
States a result. The number identifies exactly where it falls in the book — Theorem 1.1.1 is the first theorem in Section 1 of Chapter 1 — so you can refer back to it precisely later. Its proof sits right below it, folded up until you click to read it.
If $p$ is prime and $p > 2$, then $p$ is odd.
Proof 1.1.1
Suppose, for contradiction, that $p$ is even. Then $2 \mid p$, so $p$ has a positive divisor other than 1 and itself — contradicting that $p$ is prime.
Example
A worked example, titled by what it demonstrates, so you can find the one you’re after at a glance.
The sample space is the 36 ordered pairs $(i, j)$ with $i, j \in \{1, ..., 6\}$. The event “the sum is 7” contains 6 of them — $(1,6), (2,5), ..., (6,1)$ — so its probability is $6/36 = 1/6$.
Star
An observation worth keeping in mind — not a formal result (that’s a theorem) and not a mistake to avoid (that’s a warning), just something worth having front of mind going forward.
The converse is false: boundedness alone does not imply convergence — the sequence $(-1)^n$ is bounded but never converges.
Warning
A common mistake, and how to avoid making it.
Writing $a_n = L$ instead of $\lim_{n \to \infty} a_n = L$ silently claims that every term of the sequence equals $L$, not just that the sequence approaches it. The two statements are rarely both true.
Question and Answer
A numbered practice problem. The solution sits right below it, folded up until you click to reveal it — so you can try the problem yourself first.
Compute the probability of rolling a sum of 7 with two fair six-sided dice.
Solution 1
There are 6 favorable outcomes out of 36 possible outcomes, so $P(\text{sum} = 7) = 6/36 = 1/6$.