A is any finite number. Any number. as big as you want. It is saying that no matter what finite number you pick, all the terms of the sequence will exceed it eventually, after you pass some term .
An example, the sequence goes to infinity as goes to infinity.
How do we know? well, if some arbitrary number is given, all we have to do is choose any (the smallest integer greater than ) and then if , (that is, all terms after the th term will be bigger than the number .
Example, say someone choose , we can pick, , and so will all be bigger than , these will be the integers 16, 17, 18, ...
got it? No matter what number you pick, our sequence will surpass it eventually, in fact, every term of our sequence from some point on will pass it. the fact that this will happen as you keep picking bigger and bigger numbers, means the terms of our sequence can get as big as you want. in technical terms, the sequence tends to infinity
well, that would work actually--except if 0 < A < 1.
i thought getting the idea was the important thing here. do you get the idea?
to apply it, we have to be a little more technical and stick to the definition closely. you would prove this as follows:
Let be given. Choose (this is to deal with the case where 0 < A < 1, if it is in that range, we simply choose N = 1)
Then, implies .
So by definition.