In a proof of Schwarz's Lemma (Sarason, "Complex Function Theory", pp. 91-92) the function $g$ is defined in the disk by
g(z) = \begin{cases}\frac {f(z)}z&\mbox{ for } 0 < |z| < 1\\ f'(0)&\mbox{ for } z = 0.\end{cases}
The proof goes on to say that for $0 < r <1$, $g$ is bounded by $\frac {1}{r}$ on the circle $|z| = r$ and thus has the same bound in the disk $|z| \le r$ by the maximum modulus principle.
This being true for all $r$ in $(0,1)$, $g$ is bounded in absolute value by $1$.
This makes perfect sense formally.
Here are my questions:
1) Intuitively when I see something is bounded by $\frac {1}{r}$ and $r$ has values in $(0,1)$, I would naturally think of $r$ being close to $0$, and $\frac {1}{r}$ getting quite large. Then along comes the max. modulus principle and shuts it all down to a bound of $1$.
How can I think intuitively about complex analysis to see how my naive instinct does not apply here?
2) In the statement of the Lemma, we are given that $f(0) = 0$. It seems to me that this comes into play in defining $g$.
Again, how can I get an intuitive insight as to the power this stipulation has in providing the criterium for the conclusion of the Lemma.
And on a larger scale, how do I train myself to "think more complex"?
Thanks