I'm shy of curricular generalizations, but I'm gonna just go ahead and say it: Real analysis should not be anybody's *first proof-based course*. The definition of a limit has *three nested quantifiers*.
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The proper place to start with a proof-based course, though, is with a discussion of why proofs matter. For which purpose nothing is better than a collection of striking examples: http://bit.ly/2NDDF9M and http://bit.ly/2Ti7tOP provide an abundance of these.
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