I have a question, I was asked to show that $[0,1]$ and $\mathbb R$ are of equal cardinality using the Cantor-Bernstein-Schröder theorem.
I would just like some feedback, if I solved it correctly:
Let $f:[0,1]\to \mathbb R$, $f(x)=x$. it is clear to see that $f$ is injective.
Let $g:\mathbb R\to [0,1]$, $g(x)=arctan(x)+\frac{\pi}{2}$. $g$ is also injective.
Conclusion: They are of equal cardinality.
Is this correct?
The function $f$ is fine and indeed is clearly injective. The other functions you want though is a function $g:\mathbb R \to [0,1]$. You can use the $\arctan $ function but not quite in the way you did. You can easily fix this by making sure the functions you will use will be injective, have domain $\mathbb R$, and have codomain $[0,1]$.
Also, when you conclude the two sets have the same cardinality, you might want to clearly indicate that you are invoking the CSB theorem.