Let A ⊆ R be an arbitrary set of real numbers, and let K ∈ R. Prove that K is an upper bound for A if and only if the following property holds: The limit of every convergent sequence made up of elements of A is at most K.
I have no clue
Let A ⊆ R be an arbitrary set of real numbers, and let K ∈ R. Prove that K is an upper bound for A if and only if the following property holds: The limit of every convergent sequence made up of elements of A is at most K.
I have no clue
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...?