Prove or Disprove $\lim_{z \rightarrow 0} z \sin \left( \frac{1}{z} \right) = 0$.
My Approach: Since this result is true for real variable $z=x$, and since the real line contains the accumulation points of $\mathbb{C}$, by the Identity Principle the assertion holds for $z \in \mathbb{C}$ as well.
I want to see without using the "Identity Principle", how to establish the limit. Any appropriate suggestions are much appreciated.
False: Let $z=ia$ ($a$ real) then $sin({\frac{1}{z})=\frac{e^{1/a}-e^{-1/a}}{2i}}$ So $ae^{1/a}\to \infty$ for $a\to 0+$ while $ae^{-1/a}\to \infty$ for $a\to 0-$