Extended Real Number System: , and to extend the usual ordering on by declaring that for all . The completeness of can be stated as: Every subset of of has a least upper bound, or supremum, and a greatest lower bound, or infimum, which are denoted by and . If , we also write:

, .

From completeness it follows that every sequence in has a limit superior and a limit inferior:

,

The sequence converges (in ) iff these two numbers are equal (and finite), in which case its limit is their common value. One can also define and for functions , e.g. .

The arithmetical operations on can be partially extended to :

, , ,

, , .

We employ the following notation for intervals in : If ,

, , , .

If is an arbitrary set and , we define to be the supremum of its finite partial sums:

is finite $latex \}$.

**Proposition.** Given , let . If is uncountable, then . If is countably infinite, then where is any bijection and the sum on the right is an ordinary infinite series.

Proof. We have where . If is uncountable, then some must be uncountable, and for a finite subset of . If follows that . If is countably infinite, is a bijection, and , then every finite subset of is contained in some . Therefore,

.

Taking the supremum over , then

,

and then taking the supremum over , we obtain the desired result.

Some terminology concerning (extended) real-valued functions: A relation between numbers that is applied to functions is understood to hold pointwise. Thus means that for every , and is the function whose value at is . If and , is called increasing if whenever $x\le y$ and strictly increasing if whenever ; similarly for decreasing. A function is either increasing or decreasing is called monotone.

If is an increasing function, then has right- and left-hand limits at each point:

, .

is called right continuous if for all and left continuous if for all .

For points or $\mathbb C$, denotes the ordinary absolute value or modulus of , . For points or , denotes the Euclidean norm:

.

We recall that a set is open if for every , includes an interval centered at .

**Proposition. **Every open set in is countable disjoint union of open intervals.

Proof. If is open, for each consider the collection of all open intervals such that . It is easy to check that the union of any family of open intervals containing a point in common is again an open interval, and hence is an open interval. It is the largest element of . If $x, y\in U$ then either or , for otherwise would be a larger open interval that in . Thus if , the (distinct) members of are disjoint, and . For each , pick a rational number . The map thus defined is injective, for if then ; therefore is countable.

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References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 10-12.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.