I am a second year university student, my goal is to specialize in pure mathematics and for this reason I am currently trying to study calculus well through Spivak's book, and after finishing it I plan to study another book(self-study) .., and I see that the problems related to inequalities are very interesting (for Me Proving some inequalities is a fun game), so I dedicate quite a bit of time in my day to enjoy proving some of the inequalities that I find anywhere and also learn some theories related to them like (AM-AG,CBC ,holder,and schur's inequality.....), but someone told me that what I do It is nothing but a waste of time, because the inequalities are not often used in pure mathematics, frankly I have a bad feel because this person may be right, but I do not trust his words Because pure math is not his domain, so I want to be sure whether he is right or not.
so
My questions are :-
-does i have continue to develop myself in inequalities beside calculus, or do I focus only on the last subject?.
-and if the answer is yes ,What types of inequalities should I focus on?.
i know that maybe it is a bad question but it is very important for me.
any advices , comments are welcom!
Inequalities are used all the time to estimate or bound quantities in Analysis, which is a significant part of modern pure mathematics. However, there are a lot of topics in pure mathematics that don't necessarily deal with the real numbers or even an ordered field of any kind, e.g. Algebra, Category Theory, Topology, Combinatorics, Logic.
These are broad, overlapping areas of mathematics, only meant to be descriptive, not to sort the entire body of study into distinct categories. Plus, sometimes a piece of knowledge from one area gives insight into how to approach a problem in another area, even if the literal theorem does not apply. Even if a deep study in a subfield of Analysis is not in your future, you are not wasting your time by learning about how to manipulate inequalities. Keep at it!
Before you dive even deeper into this topic, I think you might enjoy this video that attempts to give you "the lay of the (mathematical) land".