<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] Since
$\quad a^3+a=b^3+b \text{ implies } (a = 0 \land b = 0) \; \displaystyle{\Large  \lor} \; \frac{a^2+1}{b^2+1} = \frac{b}{a}$
we can write, with $a \ne 0$
$$a^3+a=b^3+b \; \text{ iff } \; \frac{a^2+1}{b^2+1} = \frac{b}{a}$$
Examining the fraction you must have that both $a$ and $b$ are of the same sign.
Again, examining the fraction, assuming $0 \lt a \lt b$ leads to a contradiction.
Similarly, assuming any of
$\quad 0 \lt b \lt a\;$ or
$\quad b \lt a \lt 0\;$ or
$\quad a \lt b \lt 0$
leads to a contradiction.
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