Monthly Archives: July 2017

Proof: Every converging sequence is bounded

Posted on July 11, 2017 by Sumant Sumant

Every converging sequence is bounded. Proof: Let the sequence is converging to ie Lets choose as the sequence is converging there exists an . Find the

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Binet Formula for Fibonacci Number

Posted on July 9, 2017 by Sumant Sumant

While there are many ways to derive the recursive formula of Fibonacci Sequence. The formula to find the nth term of a Fibonacci sequence is a beautiful formula. We are not going to derive here but prove the formula using … Continue reading →

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Number of terms in Binomial, Multinomial Expansion

Posted on July 9, 2017 by Sumant Sumant

Well we know that in expansion there are terms. The question is why ? The answer is the general term of this expansion is where and if we all as one of the solution we have a total of solutions … Continue reading →

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Fibonacci Numbers through combinatorics

Posted on July 9, 2017 by Sumant Sumant

Here is one way to think about Fibonacci Numbers. 1. There are number of balls of only two colors (Let these be red and blue). 2. The red balls are identical and so are blue balls. 3. No two red … Continue reading →

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Proof of Sequence not marching to zero implies Series is not converging

Posted on July 7, 2017 by Sumant Sumant

If a sequence is not marching to zero then the series is not converging.

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Prove that a converging sequence has only one limit point

Posted on July 6, 2017 by Sumant Sumant

This is another standard theorem in introductory real analysis. There are two ways one can prove it. One by using Triangle Inequality and other by using Proof by contradiction. Given: is a sequence that converges to both and . and … Continue reading →

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Proof that e is Irrational

Posted on July 6, 2017 by Sumant Sumant

To prove that is irrational one can show that . But before one can do that one must show that is bounded above because power series is infinite. This can be done surprisingly similar to what Oresme did for the … Continue reading →

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Peano’s Axiom

Posted on July 5, 2017 by Sumant Sumant

Following are the 5 Axioms of Peano or also called as Peano’s Postulates N1. N2. If N3. is not the successor of any element in . N4. If two numbers have the same successor then N5. A subset of which … Continue reading →

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Bernoulli Inequality proof by Mathematical Induction, AM GM Inequality

Posted on July 5, 2017 by Sumant Sumant

This is the proof of Bernoulli Inequality for positive integers using Mathematical Induction This is the proof for the case when the power is between 0 and 1

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Density of Rational Numbers in Real Numbers

Posted on July 4, 2017 by Sumant Sumant

The idea is that between any two real numbers there is a rational number. To prove this we use Archimedean Property. Let .

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