Question

A sequence is said to converge to value A if, for any positive criterion c, there is an n such that all the terms in the sequence after the n-th are within a region of radius c about A.

Answer 1

@dape explain this

Answer 2

So in that picture the big blob is the set in which A is found (if A is a real number, the blob could be a interval of real numbers, for example). The thing you wrote basically says that if you give me any positive c, and I can find a n such that every successive term in the sequence can be found inside the circle with radius c, then it converges to A. In other words, you can just pick smaller and smaller c, and if it converges I will always be able to find n so that each new term in the sequence is in the smaller and smaller circles. So if you pick a c very very close to 0, the value of the sequence at n will be very very close to A.
Is that clear?