Prove that a half-open line is uncountable - corrected

Question

I have to provide a proof that a halfopen line $L_a:={𝑥 ∈ ℝ:x > a}$ 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.

Answer 1

The function $g$ doesn't make sense. For instance, $g(-1)\notin L_a$.

You can simply say that the function$$\begin{array}{ccc}\Bbb R&\longrightarrow&L_a\\x&\mapsto&a+e^x\end{array}$$is a bijection. Therefore, $L_a$ is uncountable.

Answer 2

Let $R_a:= \{x \in \mathbb R: x<a\}$ and define $f: L_a \to R_a$ by $f(x)=2a-x.$

Then $f$ is a bijection. Suppose that $L_a$ is countable, then $R_a$ is countable, hence

$$\mathbb R= L_a \cup R_a \cup \{a\}$$

is countable, a contradiction.

Answer 3

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.