<s>[INST] What is the concrete mathematical argument why $a^3+a=b^3+b\Rightarrow a=b$ is true in $\mathbb{R}$? The question arose from the following proof:
$f(x)=x^3+x+1$ is injective:
Let $a,b\in \mathbb{R}$ and $f(a)=f(b)$:
$f(a)=f(b)\Leftrightarrow a^3+a+1=b^3+b+1\Leftrightarrow a^3+a=b^3+b \Rightarrow a=b$
Is $a^3+a=b^3+b\Rightarrow a=b$ mathematically rigorous enough? I don't know the concrete mathematical argument why $a^3+a=b^3+b\Rightarrow a=b$ is true?
 [/INST] If $a^3+a=b^3+b$, then $$(a^3-b^3)+(a-b)=0 \quad \Leftrightarrow $$
$$(a-b)(a^2+ab+b^2+1)=0.$$
However, $a^2+ab+b^2+1=(a+\frac{b}{2})^2+\frac{3b^2}{4}+1>0$. This immediately implies that $a-b=0$, i.e. $a=b$. Please let me know does it make sense.
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