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{{Math-h}}

{{Math-h}}

'''Real analysis''' is a field in [[mathematics]] that focuses on the set of [[real number]]s, their properties, [[sequence]]s and [[function]]s.  Included in this branch of mathematics are the concepts of [[Limit_%28mathematics%29|limit]]s and [[convergence]], [[calculus]], and properties of real-valued functions such as [[continuous|continuity]].  It also includes [[measure theory]].

'''Real analysis''' is a field in [[mathematics]] that focuses on the set of [[real number]]s, their properties, [[sequence]]s and [[function]]s.  Included in this branch of mathematics are the concepts of [[Limit (mathematics)|limit]]s and [[convergence]], [[calculus]], and properties of real-valued functions such as [[continuous|continuity]].  It also includes [[measure theory]].

For the purposes of this article, "analysis" will be limited to the generalization and extension of the concepts of calculus, using the concepts of elementary [[point-set topology]].

For the purposes of this article, "analysis" will be limited to the generalization and extension of the concepts of calculus, using the concepts of elementary [[point-set topology]].

The reader should be quite familiar with the concepts of calculus, especially limits and continuity.  In particular, the reader should be comfortable with the ramifications of the phrase "[[Limit_%28mathematics%29|for every epsilon]]".

The reader should be quite familiar with the concepts of calculus, especially limits and continuity.  In particular, the reader should be comfortable with the ramifications of the phrase "[[Limit (mathematics)|for every epsilon]]".

==Open sets==

==Open sets==

Theorem:  Neighborhoods are open sets.

Theorem:  Neighborhoods are open sets.

Proof:  Suppose a neighborhood has center <math>C</math> and radius <math>r</math>. Let <math>p</math> be a prime number. If a point <math>x</math> is in that neighborhood, its distance from <math>C</math> must be strictly less than <math>r</math>, call it <math>k</math>. If <math>n</math> is a positive integer and <math>p</math> divides <math>n</math>, then <math>n</math> is a multiple of <math>p</math>, and therefore <math>n</math> is also in the neighbourhood of <math>p</math>.

Proof:  Suppose a neighborhood has center <math>C</math> and radius <math>r</math>. If a point <math>x</math> is in that neighborhood, its distance from <math>C</math> must be strictly less than <math>r</math>, call it <math>k</math>.

::<math>\|x-C\| = k,\ \ \ k < r\,</math>

::<math>\|x-C\| = k,\ \ \ k < r\,</math>

Place a new neighborhood, of radius <math>(r-k)/2</math>, around <math>x</math>.  Every point in that neighborhood has a distance less than <math>k + (r-k)/2</math> from <math>C</math>.  That distance is less than <math>r</math>, and this distance is a multiple of <math>p</math>, so every point in the new neighborhood is in the original neighborhood, so the new neighborhood lies within the original one.

Place a new neighborhood, of radius <math>(r-k)/2</math>, around <math>x</math>.  Every point in that neighborhood has a distance less than <math>k + (r-k)/2</math> from <math>C</math>.  That distance is less than <math>r</math>, so every point in the new neighborhood is in the original neighborhood, so the new neighborhood lies within the original one.

Theorem:  Any union of open sets, including unions of an infinite number of open sets, is an open set.

Theorem:  Any union of open sets, including unions of an infinite number of open sets, is an open set.

Closed sets are sometimes given a different definition, as sets containing their limit points.

Closed sets are sometimes given a different definition, as sets containing their limit points.

[[category:mathematics]]

*[[Bolzano-Weierstrass theorem]]

 

[[Category:Mathematics]]