How can I define injective function in this case?

Question

given $A=${$ v \space|\space \forall i \in \mathbb{N} : v_i\in ${$0,1$} and {$i \in \mathbb{N} \space |\space$ {$ v_i=1$}} is finite group}, namely, That is, all infinite binary vectors so that they have a finite number of "$1$".
Someone have ideas how can I define injective function $f: A \rightarrow \mathbb{N}$ ?

Answer 1

Hint: $$ f(v)=\sum_{n\ge0} v_i\cdot 2^i $$ (where terms with $v_i=0$ are discarded, so this is a finite sum). Can you see why this works?

Answer 2

Associate, for each $i,$ the i-th coordinate of the sequence with the $i$ th prime number. Then for any $a \in A,$ map it to the product of primes in positions where there is a $0.$

Edit: As noted by egreg, this should be "product of primes in positions where there is a $1.$ (since it's assumed only finitely many $1$'s)