I am looking for book (or any other resource) recommendations to practice "epsilonics", and technical analysis of the asymptotics of integer sequences (or real sequences), hard analytical estimates, etc.
To give an example of what I have in mind, the following is from a paper of Erdos on the number of partitions $p(n)$ of a positive integer $n$.
Let $d=\liminf \frac{np(n)}{e^{cn^{1/2}}}$, $D=\limsup \frac{np(n)}{e^{cn^{1/2}}}$, where $c=\pi\sqrt{2/3}$. Since $p(n)$ is an increasing function of $n$ there exists a $c_2$ such that for every $m$ in the range $n \le m \le n + c_2n^{1/2}$, we have $$ \frac{mp(m)}{e^{cm^{1/2}}} > \frac{D + d}{2} $$ Now we claim that for every $r_1$ there exists a $\delta_{r_1} = \delta(r_1)$ such that for $n \le m \le n + c_2n^{1/2}$, $$ \frac{mp(m)}{e^{cm^{1/2}}} > d + \delta_{r_1} $$
Then he goes on to prove the lemma.
When I see statements such as "for every $r_1$ there exists a $\delta_{r_1}...$, or "there exist constants $c$ and $x_0$ such that some function $f(n) < cn^{1/2}$", I'm in complete awe. I can read, but can't easily understand such hard technical statements, though the prerequisites seem to be nothing beyond a first course in real analysis, which I have met (I'm an undergrad student).
I want to learn how to create such technical arguments.Where does one go to get good at such things (None of my classes offer these sort of things)? What are some good resources for practicing?