I recently came across the following problem
Pick out the true statements:
There is a bijective analytic function from $\Bbb C$ to the upper half plane $\Bbb H$
There is a non-constant bounded analytic function on $\Bbb C \setminus \{0\}$
If $\{a_n\}$ and $\{b_n\}$ are two sequences of positive real numbers with $a_n \longrightarrow 0$ and $b_n$ diverging to $\infty$, then the sequence $c_n=a_ne^{ib_n} \longrightarrow 0$
My try
For the first bullet, the statement is false, since if it is true, then the range of such a function is dense in $\Bbb C$ but $\Bbb H$ is not dense in $\Bbb C$.
For the third one, the statement is true. Since $$c_n \longrightarrow 0 \iff (\Re( c_n) \longrightarrow 0) \wedge (\Im(c_n) \longrightarrow 0)$$ The above is true, since $$\vert \Re (c_n) \vert=\vert a_n \cos b_n\vert \leq \vert a_n \vert \to 0$$ and $$\vert \Im (c_n) \vert=\vert a_n \sin b_n\vert \leq \vert a_n \vert \to 0$$
Is this correct? May I have a hint for the second one?
The third one is simpler than that. Just use the fact that $\lvert a_ne^{ib_n}\rvert=a_n$.
For the second one, think about the type of singularity that such a function would have at $0$.