I agree. I think real analysis should really be taking a more topological approach to limits and continuity. In a topology classroom, they would instead define a limit in the real numbers as “every open ball around your limit point contains all of the elements of the sequence past a certain index”, which is much the same as your description of Terry Tao’s “ϵ-close” and “eventually ϵ-close”. Likewise, a continuous function would be defined, “For every open ball around f(x) in the range, there is an open ball around x in the domain where points around the domain ball get mapped inside the range’s ball.” The whole—ϵδdefinition obscures what is really going on with a bunch of mathematical jargon.
I agree. I think real analysis should really be taking a more topological approach to limits and continuity. In a topology classroom, they would instead define a limit in the real numbers as “every open ball around your limit point contains all of the elements of the sequence past a certain index”, which is much the same as your description of Terry Tao’s “ϵ-close” and “eventually ϵ-close”. Likewise, a continuous function would be defined, “For every open ball around f(x) in the range, there is an open ball around x in the domain where points around the domain ball get mapped inside the range’s ball.” The whole—ϵδdefinition obscures what is really going on with a bunch of mathematical jargon.