Can we prove that set of all nested continued fractions $\mathbb{N}^\mathbb{N}$ and the set of real numbers $\mathbb{R}$ are homeomorphic?

Question

Let us have a general sequence $(a_n)\in\mathbb{N}^\mathbb{N}$ ($a: n\in\mathbb{N}\mapsto a_n\in\mathbb{N}$).

{Question 1}

If we simply insert that sequence into the nested fractions, then we get $$a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\dots}}}\in\mathbb{R}^+$$

It seems that for any such sequence we get a converging sequence, which is a real number. So correct me if I am wrong, but that seems a straightforward bijection of all such sequences onto positive real numbers. Also, we are able to build a ordering simply in the following manner:

def compare(a,b):
    i = 0
    while (a[i]==b[i]):
        if i%2 == 0:
            if a[i]<b[i]:
                return "a > b"
            elif a[i]> b[i]:
                return "a < b"
            else:
                i += 1
        else:
            if a[i]<b[i]:
                return "a < b"
            elif a[i]> b[i]:
                return "a > b"
            else:
                i += 1    

I never saw that construction alongside others like Dedekind, Weierstrass or Cantor approaches.

{Question 2}

Have anybody investigated the topologies on $\mathbb{N}^\mathbb{N}$ for this mapping to be continous?

UPD:

I found a 1982 publication https://leopard.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00031201/Rieger_A_new_approach_to_the_real_numbers.pdf

Rieger, Georg Johann A new approach to the real numbers (motivated by continued fractions) Veröffentlicht in: Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft Band 33, 1982, S.205-217

UPD2:

The main problem of the definition above is $0$, since $1=0+\frac{1}{1}$. So one way or another we need to factorize $\mathbb{N}^\mathbb{N}$.

UPD3:

It seems that the correspondance is more complicated, like $$(\mathbb{Z}×\mathbb{N}^\mathbb{N})\cup (\bigcup_{n\in\mathbb{N}}\bigcup_{(a_k)\in\mathbb{Z}\times\mathbb{N}^n}\{( a_0,a_1,a_2,\dots, a_n,0,0,0,\dots)\}\sim\mathbb{R}$$