TITLE: How important are inequalities?
QUESTION [7 upvotes]: When reading the prefaces of many books devoted to the theory of inequalities, I found one thing repeatedly stated: Inequalities are used in all branches of mathematics. But seriously, how important are they? Having finished a standard freshman course in calculus, I have hardly ever used even the most renowned inequalities like the Cauchy-Schwarz inequality. I know that this is due to the fact that I have not yet delved into the field of more advanced mathematics, so I would like to know just how important they are. While these inequalities are usually concerned with a finite set of numbers, I guess they must be generalised to fit into subjects like analysis. Can you provide some examples to illustrate how inequalities are used in more advanced mathematics?

REPLY [3 votes]: I think they're most important because of limits. I'm sure you've done limits in your calculus class. Limits are extremely important in maths - they're not just used to define derivatives and integrals.  
There's a whole branch of mathematics called analysis which deals with limits. Sometime in the 18th century, mathematicians tried to understand calculus more rigorously and they came up with a formal definition of a limit.  
$\lim \limits_{n \rightarrow \infty} a_n = a \iff \forall \varepsilon > 0, \exists N \in \mathbb{N} \ s.t. |a_n - a| < \varepsilon $ 
(In simple words, if you have a sequence of numbers say $ \dfrac{1}{2}, \dfrac{3}{4}, \dfrac{7}{8}, \dfrac{15}{16}...$ which tend to a number (1 in the previous example), after a certain point ($N$) all points within a certain range ($\varepsilon$) of the limit.)
Because of this, analysis is all about inequalities - including the triangle inequality and the Cauchy-Schwartz.  They're very useful.
There are other places such as in Computer Science, when you define the order of growth of an algorithm (Big-O notation), and Operations Research where you use them to put certain constraints on maximisation/minimisation problems, e.g. find the best portfolio of investments, given that you may at most invest $1000.  (The last one requires a good knowledge of theoretical probability - which is basically analysis and inequalities.)