Suppose that is a sequence of real numbers. Define a sequence by
(a) Prove that converges to a real number if converges to .
(b) If converges, does also converge?
Suppose that is a sequence of real numbers. Define a sequence by
(a) Prove that converges to a real number if converges to .
(b) If converges, does also converge?
I think that (a) can be proven fairly easy within the definition of convergent sequences.
(b) on the other hand I think is false. A possible counter example is
Suppose that is a sequence of real numbers. Define a sequence by
(a) Prove that converges to a real number if converges to .
(b) If converges, does also converge?
For the first one we can easily prove that if then
Proof: We know that since we may choose such that then and since we must have that
This implies that and consequently exist. So because of this existence