Assume $A$ is a countable subset of $\mathbb{R}$ and that $x$ is a real number. Prove that if we shift each element in $A$ by $x$, this new shifted set is also countable.
Okay, I think I'm on the right track but it's hard for me to tell for sure.
$\textbf{Proof:}$ Assume $A$ is a countable subset of $\mathbb{R}$. Because $A$ is countable set then $A$ must be bijective.
This is pretty much as far as I got. Shifting the elements in $A$ by $x$ would mean $A+x$ right? If that's the case then I'd have to prove this new set is also bijective. However, not sure how to go about doing that.
Let$$B = \{a + x | a \in A\}$$ be the shifted set and because $A$ is countable there exists a bijection between $\Bbb N$ and$A$ let's call it $$g: \Bbb N \to A$$
Now define a new function $f$ by $$f(n) := g(n) + x $$ and show that $f$ is a bjective mapping from $\Bbb N \to B$.
So we've found a bijection between $\Bbb N$ and $B$ hence $B$ is countable.