Let $A$ be a set of sequence starting with $0$ followed by an infinite number of terms
Let $B$ be a set of sequences whose last term term is $1$ preceded by an infinite number of terms
Question 1: Is A intersection B empty?
Question 2: Is it possible/reasonable to define C, directly as a set of sequences that start with $0$ and have the last term $1$ with an infinite number of terms?
All I know is that A obviously makes sense, and B makes sense if you start defining it from the end and go backwards. I'm not even sure if Question 1 and Question 2 are equivalent.
I don't think your definition of $B$ is mathematically corret. If a sequence is preceeded by an infinite number of terms, it doesn't make sense to say that it has a last term. Something that's intuitively equivalent (in some sense) to $B$ is "The set of all sequences that converge to $1$".
With respect to this re-definition, of course $A\cap B\neq \phi$ as the sequence $0,0+\frac12,0+\frac12+\frac14,\dots$ is in $A\cap B$.
Also, it makes perfect sense to define $C$ as the set of all sequences that start with $0$ and converge at $1$.
Does that help?