Proof: The Rationals are Dense in the Reals | Real Analysis

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Between any two real numbers there exists a rational number. This is what is meant by "the rationals are dense in the reals". Take two real numbers and we can find a rational in between them, a very handy thing! We prove this with the Archimedean Property in today's real analysis video lesson. #realanalysis

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The proof is a bit misleading as it uses an additional property of the integers that is not mentioned here. Namely that for any real number x there exists a minimal integer k such that k - 1 <= x < k. This is a consequence of the Archimedean property and the well-ordering on the natural numbers.

xppe

oh my goodness, you filmed this at 2 am? Thanks for your hard work XD

jinglepinglepie

This is an abstract and difficult concept . However, it has been so clearly explained in this great video .Thank you for your great work ❤

wtt

Hey, I love you. Continue the good work.

mandeath

can we chose m=a-b and n=2 and work from there?

Jancel

Great video. Is any course available for " Foundaton of Mathematics"

sugathmudalige

if anyone's curious about the irrationals
if a and b are irrational, atleast one of (a+b)/2 or (a+3b)/4 will be irrational, if both were rational, then (a+3b)/2 would be rational and (a+3b)/2 - (a+b)/2 = 2b/2 = b would be rational, which wouldn't make sense
if only one of them is irrational, just do (a+b)/2
if both are rational, do (a + pi(b))/(1+pi), since that is just equal to b - (b-a)/(1+pi) and b-a isn't zero, it's irrational

lox

Is there a video on dedekind cuts and their use in forming the reals?

priyankaparikh

it's a great video thumbs up. if i may ask, y do u have replace "a" by "b-1/n"

AblieFatty-pbds

One would have to be dense not to be a fan of Wrath of Math! 😂

punditgi

if x is a real number, prove that -x≤|x| can you prove this

sumitlama

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