Question

a) give an example to show it is possible to have lim(x_sub(n) + y_sub(n)) exist without having lim x_sub(n) or lim y_sub(n) exist

b) give an example to show it is possible to have lim(x_sub(n) * y_sub(n)) exist without having lim x_sub(n) or lim y_sub(n) exist

Answer 1

Question 1. (a) Give an example to show it is possible to have $\lim(x_n + y_n)$ exist without having $\lim x_n$ or $\lim y_n$ exist.

(b) Give an example to show it is possible to have $\lim(x_n y_n)$ exist without having $\lim x_n$ or $\lim y_n$ exist.

Solution. (a) Consider $x_n = (-1)^n$, $y_n = -x_n$. Then $x_n + y_n = 0$ for all $n \in \mathbb{N}$, so $\lim(x_n + y_n) = 0$. But $\lim x_n$ does not exist, because there are two subsequences $x_{2n} = 1$ and $x_{2n-1} = -1$, which have different limits (1 and -1, respectively). Similarly $y_n$ does not have a limit.

(b) Take $x_n = y_n = (-1)^n$. As was shown above, this sequence does not have a limit. Nevertheless $x_n y_n = (-1)^{2n} = 1$ for all $n$, therefore, $\lim(x_n y_n) = 1$.