1. The problem statement, all variables and given/known data I'm wondering why we can't use less than or equal to for the formal definition of the limit of a function: 2. Relevant equations lim x→y f(x)=L iff For all ε>0 exists δ>0 such that abs(x-y)<δ implies abs(f(x) - L)<ε Why not: lim x→y f(x)=L iff For all ε>0 exists δ>0 such that abs(x-y)=<δ implies abs(f(x) - L)=<ε 3. The attempt at a solution I just imagine that it doesn't matter because epsilon is arbitrary, but I'm not sure. If we're using the epsilon such that we have a strict inequality, given that set of x that satisfies this, all we're doing is including the boundary points of the open set specified by delta on the domain, and we know that by the definition using strict inequalities that there then exists a [ε][/1]>ε such that abs(f(x) - L)<[ε][/1] (and thus abs(f(x) - L)=<[ε][/1], but we know that there are not x in our ball that actually satisfy abs(f(x) - L)=[ε][/1]). But I can't seem to find a way to satisfy a two way implication i.e. prove the definitions are equivalent. Maybe there's a good contradiction up someone knows two show they're not equivalent?