Having read several undergraduate textbooks in complex analysis (Stein-Shakarchi, Gamelin, etc.), I find that some "hand-waving" arguments are frequently used. An example (the proof of the Cauchy integral formula in Stein-Shakarchi) is given in the appendix regarding what I mean for "hand-waving".
Hand-waving arguments in students' homework may be sometimes (usually?) due to a certain degree of "dishonesty": one wants to gloss over a part that one does not fully understand to get credits. If one is brutally honest to oneself when doing homework, then one might not use such arguments at all or one would explicitly admit that one is using such arguments so that the readers should be cautious in advance.
On the other hand, textbooks written by masters are another story.
What are possible reasons for making hand-waving arguments in textbooks of undergraduate analysis? (Some are deliberately left as helpful exercises for the readers but some are not. I should be referring to those non-exercise ones.) A hand-waving argument in algebra is very different from one in analysis and I would like to restrict my question in the context of analysis here.
In the views of students, how should one deal with such arguments for learning?
[An example of hand-waving arguments in Stein-Shakarchi.]