proof of Cauchy condition for limit of function

The forward direction is . Assume that limxx0f(x)=L. Then given ϵ there is a δ such that

|f(u)-L|<ϵ/2 when 0<|u-x0|<δ.

Now for 0<|u-x0|<δ and 0<|v-x0|<δ we have

|f(u)-L|<ϵ/2 and |f(v)-L|<ϵ/2

and so


We prove the reverse by contradictionMathworldPlanetmathPlanetmath. Assume that the condition holds. Now suppose that limxx0f(x) does not exist. This means that for any l and any ϵ sufficiently small then for any δ>0 there is xl such that 0<|xl-x0|<δand|f(xl)-l|ϵ. For any such ϵ choose u such that 0<|u-x0|<δ and put l=f(v) then substituting in the condition with u=xl we get |f(xl)-l|<ϵ. A contradiction.

Title proof of Cauchy condition for limit of function
Canonical name ProofOfCauchyConditionForLimitOfFunction
Date of creation 2013-03-22 18:59:08
Last modified on 2013-03-22 18:59:08
Owner puff (4175)
Last modified by puff (4175)
Numerical id 8
Author puff (4175)
Entry type Proof
Classification msc 54E35
Classification msc 26A06
Classification msc 26B12