Let {Xn}, {yn} be real sequences and b,x,y be real numbers.
Prove (using limit theorems)
If Xn goes to X and Yn goes to Y as n goes to infinity, then
i)lim (Xn + Yn) = X+Y
ii)lim (bXn) = bX
iii) lim (XnYn) = XY
(all limits n going to infinity)
Thanks
The second one is really easy try that one yourself. This one is the hardest of all of them.
Lemma: Convergent sequences are bounded.
Proof: Let be a convergent sequence then for we have for . Thus, . This means are all bounded by . So let then it means for all . Q.E.D.
Now we can prove your problem.
We know that the sequence is bounded by some positive number .
This means,
.
Thus,
.
This means,
.
Q.E.D.