On studying equivalent (or not) definitions; the case of limits in R and R 2

Abstract

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the ϵ − δ one for limits of one-variable real functions that are based on the realisation of the interval of δ ’s. In the second one, students are directed by their instructor to restore equivalence between two distinct definitions of limit for a function mapping a subset of R 2 into R . This is done by adding a condition to one of the definitions, resulting in a third definition that is equivalent to the other one.