A sequence is an endless list of numbers in a fixed order. Each place in the list has a number, and the same number may appear at many places. Order and repetition both matter, which is what separates a sequence from the set of the values it takes.

A sequence is a diary, not a wardrobe. A wardrobe holds a set of coats; a diary records which coat was worn on each day, in order, and the same coat may appear on many days.

A term standing at every place, and the same value may stand at many of them: 1 is the term at every odd index here. That is what separates the sequence from the set $\{1,2,3,4,5\}$ of values it takes — a set forgets both the order and the repetition, and a sequence is made of them.

A sequence gives a real number for each natural number. The number at place n is written with a subscript, and the list as a whole is written in brackets. Everything in this chapter is a statement about all places from some point onwards.

• $(a_n)$ is bounded above when some $M$ satisfies $a_n \le M$ for every , and bounded below in the mirror way.

There is a term at every place, so infinitely many terms, but only two values appear.

$a_n = \tfrac{1}{n}$ is bounded above by $1$.

The first term is $1$ and the rest are smaller, so no term exceeds $1$.

$a_n = \tfrac{1}{n}$ is decreasing.

Each term is smaller than the one before, at every step without exception.

If $a_n = n^2$ then $a_3 = 9$.

The third place holds the square of three.

Its domain is the natural numbers, and the subscript is only notation.

• A sequence is a function on $\mathbb{N}$, and its set of values forgets order and repetition.
• Bounded above, bounded below, increasing, decreasing and monotone are each read off the terms.
• A sequence may be given by a formula or by a recursion.
• Eventually means from some place onwards, which is all a limit can see.

Great work! You now know how to:

• Read a sequence as a function on the natural numbers
• Compute terms from a formula and from a recursion
• Tell terms from values, and say what eventually means