How can I use these two bijections to form a bijection $\mathbb{R}^{\mathbb{N}} \to \mathbb{R}$? Build a bijection $\mathbb{R}^{\mathbb{N}} \to \mathbb{R}$  by using the two following known biections $\varphi:{\mathbb{N}} \to {\mathbb{N}} \times {\mathbb{N}}$ and $\psi:{\mathbb{R}}  \to \{0,1\}^{{\mathbb{N}}}$.
Edit.
My solution. 
Use  the classical bijection $\varphi: \mathbb{R}^2 \to \mathbb{R}.$ Now construct a bijection  $\Phi: \mathbb{R}^{\mathbb{N}} \to \mathbb{R}$  by 
$$
\Phi(x_1,x_2,\ldots,x_n, \ldots)=\varphi(...\varphi(\varphi(x_1,x_2),x_3)...)
$$
I know  that it doesnt use those two proposed bijections but 
is it   correct?

Hint: Note that $\mathbb R^\omega=(2^\omega)^\omega=2^{\omega^2}=2^\omega=\mathbb R$. If you can find a bijection for each equality, composing them should give you your desired bijection.
