Why is it not possible to construct a finite state machine that recognizes precisely those sequences in the language
where the alphabet for A is {0,1}..
I just don't get it why this is not possible...Maybe I am not seeing something as I am new to this.
The language you posted is not regular, only regular languages (i.e. defined by a regular grammar) can be accepted by a finite state machine.
The reason for this is, non formally, that finite automata can not count because they have a finite number of states. This would be required for comparing i to j in your example.
The construct that will be able to accept your language would be a stackautomaton, because your language is contextfree. See the Wikipedia article about chompsky hierarchy1 for additional details.