At project-perelman, We have inevitably started delving into Cauchy sequences and real analysis because if you really want to believe calculus proofs and have enough genuine interest and analytical mind, you will discover very fast that there are missing foundations and hand-waving proofs all over the place if you do not go deeper. This is best expressed by Karl Hahn in his excellent website.

As the first thing in introducing calculus he says:

As an example, my (failed) proof (http://jadnohra.net/release/math/evt.pdf) of the Extreme Value Theorem does not perfectly work because I could not prove that my induction would cover the whole space of inputs. This can only be done by digging deeper than calculus.

That page led me to yet another page by Eric Schechter, where I found a quote I really like:

*"*

*There is a natural way to "add" or "multiply" two points in the Euclidean plane. By "natural" I mean that the definitions have turned out to be useful for many applications, and that the definitions are fairly simple"*

It puts in words what we many times feel but cannot express, when something mathematical feels 'natural'. Worth memorizing!

To all of this, I will add this unrelated and brilliant quote that my lovely wife just sent me:

".

*..höre nie auf zu zweifeln. Wenn Du keine Zweifel mehr hast, dann nur , weil du auf deinem Weg stehen geblieben bist. .... Aber achte auf eines: Lass nie zu, dass Zweifel dein Handeln lähmen. Treffe auch dann immer die notwendigen Entscheidungen, wenn du nicht sicher bist, ob deine Entscheidung richtig ist. .... Paolo Coelho, Brida*"
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