I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt:
Is the set of all irrational real numbers countable? Prove your answer.
My Proof:
Suppose the irrational reals were countable. Then the union of the rational reals (which we know to be countable) and the irrational reals would be countable, as the union of two countable sets is countable. However, this would then mean that the reals were countable, which we know to be false. Thus, the irrational reals must not be countable.