Here's a start in each direction:

Suppose there's no sequence a_n in A with limit L>K. Since for any a in A, the sequence {a,a,a,...} has limit a, we know...?

Now suppose there is a sequence a_n in A with limit L>K. Let epsilon=L-K. By the limit definition of a sequence, there must be an element of the sequence closer to L than epsilon, which means...?