Prove that a half-open line is uncountable - corrected I have to provide a proof that a halfopen line $L_a:={ ∈ ℝ:x > a}$ is uncountable.
I have used Schröder-Bernstein-Cantor and tried to show that an injection exists between $L_a$ and $ℝ$ an injection between $ℝ$ and $L_a$ to conclude that a bijective function exists between $ℝ$ and $L_a$ to prove that $L_a$ is uncountable.
I chose the following two functions:
$f:L_a \rightarrow ℝ:x \rightarrow x$ is injective
$g:ℝ \rightarrow L_a:x \rightarrow a+e^x$ is injective
It follows that a bijection exists between both sets, so $L_a$ is uncountable.
EDIT: corrected use of SBC, it should now work.
Thanks for your feedback.

An interval is uncountable:
$$\tan:(-\pi/2,\pi/2)\to\Bbb R$$ is a bijection.
But any two (finite) open intervals are isomorphic (there's a bijection between them): $(c,d)\leftrightarrow(a,b)$ by $$x\to \dfrac{(d-x)}{(d-c)}a+\dfrac{(c-x)}{(c-d)}b$$.
Thus $L_a$ is uncountable, because it contains an interval.
