<s>[INST] Showing that $\left|\frac{z}{z^3+1}\right|\leq\frac{R}{R^3-1}$ 
I am trying to show that if $|z|=R>1$, then 
  $$\left|\frac{z}{z^3+1}\right|\leq\frac{R}{R^3-1}$$

I've been playing around with the above inequality and this is where I am at.
\begin{align}
\text{Consider} \ \left|z^3+1\right|&\leq\left|z^3\right|+1  \ \ \ \ \ \text{(by the triangle inequality)}\\ 
&=\left|z\right|^3+1 \\
&=R^3+1 \\ \\
\Rightarrow \left|z^3+1\right|&\leq R^3+1 \\
\frac{1}{\left|z^3+1\right|}&\geq\frac{1}{R^3+1} \\
\frac{|z|}{\left|z^3+1\right|}&\geq\frac{|z|}{R^3+1} \ \ \ \ \ \ \text{(inequality unchanged, |z|>1)} \\
\left|\frac{z}{z^3+1}\right|&\geq\frac{R}{R^3+1} \\
\end{align}
I'm unsure of how to yield the desired inequality. Have I made a mistake somewhere? 
EDIT
If I used the inequality $$|z_1+z_2|\geq |z_1|-|z_2|$$ I believe this will work. My question is, does this inequality hold for real numbers as well?
 [/INST] WLOG $z=R(\cos t+i\sin t)$ where $t$ is real
$$|z^3+1|^2=R^6+1+2R^3\cos3t\ge R^6+1-2R^3= (R^3-1)^2$$
as $\cos3t\ge-1$ and $R^3>1$
</s>