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Mathematical proof

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+#REDIRECT [[Mathematical proof]]
-In analytical [[calculus]] (often times known as '''advanced calculus''', a specific subset of '''[[mathematical analysis]]'''), there are three important methods to determine that a given hypothesis is true or false.
-==Technique #1: Deductive Proof==
-Deductive proof, or [[direct proof]], is very straightforward.
-Given some hypothesis:
-There is a logical method to prove that the hypothesis is true through step by step operations and by using postulates, lemmas, and definitions in constructing the final conclusion.
-Many mathematical hypothesis can be proven using deductive proof techniques.
-'''Example'''
-:<math>If\ a \in R \ge 0</math>
-:<math>Prove\ </math> <math>a^2 \ge 0</math>
-:'''Proof:'''
-:Since <math>a \ge 0</math> then by multiplicative property of real number system,
-:<math>a \star a \ge a \star 0 </math>
-:<math>a^2 \ge 0 \mathcal{p}</math>
-==Technique #2: Proof By Contradiction==
-Given some hypothesis:
-It is automatically assumed that the conclusion is wrong, and an antihypothesis must be proven. Then, deductive proof is used (stepwise operations and definitions) to reach a final conclusion. If this final conclusion contradicts the original hypothesis then it is said that the original conditions are proven to be true and correct.
-More difficult mathematical hypothesis can be proved or disproved using [[proof by contradiction]] techniques.
-'''Example'''
-Extreme Value Theorem (Proof by contradiction):
-:<math>Assume\ f(x)\ is\ UNbounded\ on\ [a,b]\ </math>
-:<math>(i.e.\ there\ exists\ {x_n}\subset[a,b]\ such\ that\ f(x_n)\rightarrow\infty.)</math>
-:<math>Since\ x_n\ is\ bounded,\ then\ Bolzano-Weirstrauss\ Theorem\ says:\ </math>
-:<math>There\ exists\ {x_{nk}}\rightarrow C \in [a,b]</math>
-:<math>such\ that\ \lim_{n \to \infty}x _{nk} \rightarrow c,\ and\ \lim_{n \to \infty}f (x _{nk}) \rightarrow  \infty.</math>
-:<math>Since\ f\ is\ continuous\ at\ point\ C,\ then\ f(C)\ =\ \lim_{k \to \infty}f(x_k)\ =\ \infty.</math>
-:<math>Because\ x _{nk}\ is\ bounded,\ and\ f(C)\ is\ bounded,\ this\ conclusion\ is\ wrong.</math>
-:<math>Therefore\ f(x)\ IS\ BOUNDED\ on\ [a,b].\ \mathcal{p}</math>
-==Technique #3: Mathematical Induction==
-[[Mathematical induction]] is very powerful two step process, and can only be used for the natural number <math>n\ </math>
-:<math>Prove\ \sum_{k=1}^n\ 4k\ = \ 2(n+1)\ for\ all\ k\ and\ n.</math>
-:<math>Step\ 1:\ Consider\ case\ k\ =\ 1.</math>
-:Right hand side equals 4.
-:Left hand side equals 4.
-:Equation satisfied for n = 1.
-:Assume equation true for some n.
-:<math>Step\ 2:\ Consider\ case\ n+1.</math>
-:<math>\sum_{k=1}^n\ 4k\ = \ 2(n+1)\ becomes\ \sum_{k=1}^n\ 2n(n+1)\ +\ 4(n+1) = \ 2(n+1)\ \star\ 2(n+1)(n+1+1)</math>
-:Simplify left and right hand sides.
-:Left hand side equals 2(n+1)(n+2)
-:Right hand side equals 2(n+1)(n+2)
-:Both sides are equivalent.
-:Based on cases n=1 and n=n+1, mathematical induction tells us
-:equation true for all k,n, in N. <math>\mathcal{p}</math>
-==See Also==
-# [[proof theory]]
-# [[Mathematical proof]]
-==For Further Reference / Selected Bibliography==
-# Wade, William R. ''An Introduction to Analysis.'' Upper Saddle River, New Jersey: Pearson Prentice Hall, 2004.
-[[Category:Mathematical analysis]]