thats true, but then there is a dichotomy here. On the one hand, u can be crazy good @ manupilating stuff like e^x, power series, etc and not know the convergence ideas behind it, like sequence of functions, uniform convergence etc (or alternatively, defining e^x using LUB axiom, or even more alternatively, defining e^x as that unique function that solves a particular DE with a given set of boundary conditions), or u can know these things, but be less agile with calculations

due to maybe, lack of practice / drill problems. most math students in pure stream lack drill problems practice. So …

I have another example. Perhaps one learns linear algebra from say, a matrix oriented book- a la, gilbert strang. He may come across the problem: if two systems of linear equations have the same solution space, they are row equivalent. He breaks his head on this for a while, does matrix calculations cuz thats all he can do, and gives up. Then he reads sheldon axler, where the development of vector space theory and transofrmatons theory is well fleshed out that, this theorem simply

is a sub case of a general theorem holding for transformations. Id say, without actually defining the word "un…