Prove the following homeomorphism: $ R_{ \geqslant 0} \times R_{ \geqslant 0} \, \cong \,R_{ \geqslant 0} \times R $
where R are the real numbers and $ R_{ \geqslant 0} = \left\{ {x \in R:x \geqslant 0} \right\} $
If $ \,R_{ \geqslant 0} \, \cong \,\,R $ it´s done , but I think that this is not true, since it´s obvious that a continuous bijective function $ f:A \subset R \to R $ must be strictly increasing or strictly decreasing, so considering $ f(0) $ it´s easy to see that a continuous injective function between this two sets cannot be surjective, so there are not homeomorphic, but does not imply necesarly that the product it´s not. What can I do?