Addition of Alephs

Question

Prove: $\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} $

The textbook I am using has a long proof done by transfinite induction. I am looking for a direct proof.

Can I do this:

$\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} \cdot \aleph_{\alpha}$

$= \text{max} \{\aleph_{\alpha}, \aleph_{\alpha} \}$

$= \aleph_{max \{\alpha,\alpha \} }$

$= \aleph_{\alpha} $

Answer 1

HINT:

Let $\omega_\alpha$ be the least ordinal of cardinality $\aleph_\alpha$. Note that every ordinal can be written as $\delta+n$ where $\delta$ is a limit ordinal. The maps $\delta+n\mapsto\delta+2n$ and $\delta+n\mapsto\delta+2n+1$ are both injections from $\omega_\alpha$ into itself, and their ranges are disjoint.