Some questions regarding complex analysis

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 $$0$$.
• In the third one, we cannot use the divergence of $b_n$ . Is this divergence assumption, not needed here ? – Chinnapparaj R Feb 10 at 16:34