Also a few other equations related to this equation are often studied. (Equations which can be easily transformed to Cauchy functional equation or can be solved by using similar methods.) Is there some overview of basic facts about Cauchy equation and related functional equations - preferably available online?
My question is related with the definition of Cauchy sequence As we know that a sequence $(x_n)$ of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer ...
Very good proof. Indeed, if a sequence is convergent, then it is Cauchy (it can't be not Cauchy, you have just proved that!). However, the converse is not true: A space where all Cauchy sequences are convergent, is called a complete space. In complete spaces, Cauchy property is equivalent to convergence. However (for example Riemann integrable functions on $\mathbb R$, say) incomplete spaces ...
However, questions like this one make me understand that the $2^ {-n}$ condition is necessary for this to be a true statement. So I am wondering how to appeal to the Cauchy definition for this proof. Do I prove that every convergent sequence is therefore Cauchy, and then try to prove convergence?
You know that every convergent sequence is a Cauchy sequence (it is immediate regarding to the definitions of both a Cauchy sequence and a convergent sequence). Your demonstration of the convergence of the sequence is right, so yes, you have proven that $ (\vert a_n\vert)$ is a Cauchy sequence.
Cauchy-Schwarz inequality in this case is just a simple consequence of solving the least square problem mint∈Rf(t) min t ∈ R f (t). This is not "visual", but arguably very intuitive and elegant.
Since you asked specifically how to understand Cauchy sequences "intuitively" (rather than how to do $\epsilon,\delta$ proofs with them), I would say that the best way to understand them is as Cauchy himself might have understood them.
