Okay, the parts you bolded were the definition of a sequence, the definition of a limit, and the proof that the limit of the sequence of partial sums of the series represented by .999... is 1 (don't worry if you didn't understand that last sentence right now.)

First, what is a sequence of real numbers?

A

**sequence** is a list of numbers indexed by the natural numbers (1, 2, 3, 4, 5, etc.), or equivalently it can be thought of as a function from natural numbers to real numbers. Often a sequence is written x

_{1}, x

_{2}, x

_{3}, x

_{4}, ... meaning that the first number in the sequence is x

_{1}, the second is x

_{2}, etc. The numbers in the sequence are called the

**terms** of the sequence; for instance, the

*n*th term of the sequence x

_{1}, x

_{2}, x

_{3}, ... is the number x

_{n}. (You can also talk about sequences of things other than real numbers, such as sequences of complex numbers, or sequences of points in the plane, etc., but for our purposes all we need are sequences of real numbers.)

Some example sequences:

The constant sequence zero: 0, 0, 0, 0, ...

The sequence 1/

*n*: 1, 1/2, 1/3, 1/4, 1/5, ...

The sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

The sequence of powers of 1/10: 1/10, 1/100, 1/1000, ...

etc.

Second, what is a convergent sequence, and what is a limit?

A sequence of real numbers x

_{1}, x

_{2}, x

_{3}, ... is said to

**converge** to a

**limit** *l* if for every real number ε greater than zero, there is a natural number

*N* such that for every

*n*>

*N*, the

*distance* |x

_{n}-

*l*| between the

*n*th term of the sequence and the limit is smaller than ε. Roughly speaking, a sequence converges to

*l* if the points on the number line defined by the terms of the sequence eventually get as close to

*l* as you want.

Examples:

The constant sequence zero: 0, 0, 0, 0, ... converges to 0, because it is always as close as you could possibly want to zero.

The sequence 1/

*n*: 1, 1/2, 1/3, 1/4, 1/5, ... converges to 0, because given ε>0, we can consider the real number 1/ε. There is some natural number

*N* larger than 1/ε (this is sometimes called the Archimedian property of the real numbers). Then for

*n*>

*N*, 1/

*n*<1/

*N*<ε, so |1/

*n*-0|<ε. This is exactly what it means for the sequence to converge to 0.

Third, what is an infinite sum?

An infinite sum, usually called an infinite

**series**, is simply the sum of an infinite number of terms arranged in a sequence, and is defined by means of limits. If the terms of the series are y

_{1}, y

_{2}, y

_{3}, ..., then we define the

*n*th partial sum s

_{n} of the series as the finite sum y

_{1}+y

_{2}+y

_{3}+...+y

_{n}. The series is said to

**converge** to a limit s if the sequence of partial sums s

_{1}, s

_{2}, s

_{3}, ... converges to s (as defined above for a sequence), in which case we write y

_{1}+y

_{2}+y

_{3}+...=s.

Now does the proof make some sense? (By the way, can you see this symbol: ε? I want to make sure it displays properly on your browser. It's the lower case Greek letter epsilon, and it is almost always used as I have used it here in the context of limits in mathematics.)

p.s. The contents of this post constitute material which is not generally taught except in college math courses, and some high school calculus classes.

Edit: I decided to define infinite series in more detail as well.

You can also check out the Wikipedia pages on

limits (of sequences) and

series.