$c$ is the cardinal of real numbers, prove $c+c=c$.

Question

$c$ is the cardinality of real numbers. To prove that $c+c=c$ we need to show that $c+c$ is less or equal to $c$ and vice versa. Since $c$ is a cardinal number I choose $(0,1)$ and $(1,2)$ (separate sets) as the sets where their cardinals are $c$. Now where do I go? I know that $(0,1)\sim (a,b)$ and maybe this can help to complete the proof. Can anyone give a hint??

Answer 1

Since $(0,1) \subset (0,1)\cup(0,2) \subset (0,2) \sim (0,1)$ we have $c \leq c+c \leq c,$ so $c+c=c.$