For starters, if a sequence is unbounded, then it is not convergent. If it converged to, say, a number L, then eventually all the entries of the sequence would fall within (L-1, L+1). (This is like picking epsilon = 1.) There would be finitely many numbers that DIDN'T fall in that range. So, you've got a handful of numbers, possibly large, but only finitely many, so there's a biggest. And the rest are sandwiched between L-1 and L+1, so they are bounded. Thus the whole sequence is bounded.
What that means is, a sequence can never be unbounded and convergent. (which rules out #3)
So: if you can prove it converges, you automatically know it is bounded. And if you can prove is it unbounded, you automatically know it is not convergent.
The sequence 1/An is neither convergent, nor bounded.
To see this, we first try an example.
Consider: {An} = 1, 1/2, 1/3, 1/4, 1/5, ...
Then, the sequence {1/An} is 1,2,3,4,5,...
This sequence is unbounded.
Now, for the details:
If An converges to 0, then eventually all the numbers are within 1 of 0. That is, |An|<1
But that means that |1/An|>1
So eventually, all of the numbers in 1/An are larger than 1 or less than -1
We can repeat this argument:
If An converges to 0, then eventually all the numbers are within 1/2 of 0. That is, |An|<1/2
But that means that |1/An|>2
So eventually, all of the numbers in 1/An are larger than 2 or less than -2
If An converges to 0, then eventually all the numbers are within 1/10 of 0. That is, |An|<1/10
But that means that |1/An|>10
So eventually, all of the numbers in 1/An are larger than 10 or less than -10
We can play this game for any number.
Thus, we can show that, eventually, all of the numbers in the sequence |1/An| are very large, as large as we would want. Thus, the sequence is unbounded.
Thus, it can't be convergent.
PS:
In the example An=1, 1/2, 1/3, 1/4, ...
1/An = 1, 2, 3, 4, 5...
The sequence 1/An "diverges to infinity"
So, the sequence still has a limit, but the limit is infinity.
Even THAT doesn't always happen:
Consider:
An = 1, -1/2, 1/3, -1/4, 1/5, ...
Then 1/An = 1, -2, 3, -4, 5, -6
so the numbers aren't getting "closer and closer to infinity"
they are bouncing around while getting larger and larger in magnitude