elementary set theory - Is there a bijection between $mathbb{N} times mathbb{R}$ and $mathbb{R}$

So I'm doing some practice on set theory, and I am having some trouble proving a lemma.

Basically I want to ask if there is there a bijection between $\mathbb{N} \times \mathbb{R}$ and $\mathbb{R}$

If yes, could someone provide a simple construction of such a bijection?

Any help or insights is deeply appreciated.

Answer

For just answering the yes/no question, the easiest way is to use the Swiss knife of bijections, the Cantor-Schröder-Bernstein theorem, which just requires us to construct separate injections in each direction $\mathbb R\to\mathbb N\times\mathbb R$ and $\mathbb N\times\mathbb R\to\mathbb R$ -- which is easy:

$$ f(x) = (1,x) $$

$$ g(n,x) = n\cdot \pi + \arctan(x) $$

Because there is an injection either way, Cantor-Schröder-Bernstein concludes that a bijection $\mathbb R\to\mathbb N\times\mathbb R$ must exist.

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If you already know $|\mathbb R\times\mathbb R|=|\mathbb R|$, you can get by even quicker by restricting your known injection $\mathbb R\times\mathbb R\to \mathbb R$ to the smaller domain $\mathbb N\times\mathbb R\to\mathbb R$ instead of mucking around with arctangents.