If $s_n$ is an oscillating unbounded sequence and $t^n$ is a bounded sequence then $s_n + t_n$ is oscillating and unbounded?

Question

Let $\{s_n\}$be an oscillating sequence and not bounded and $\{t_n\}$ be a bounded sequence. Then $\{s_n + t_n\}$ is oscillating and unbounded. True or not?

Since $\{s_n\}$ is unbounded the sequence $\{s_n\}$ does not converge and also does not diverge to $ \infty $ or $- \infty $, then it is similar to $\{(-1)^n n\}$.

For this I can say that the sum $\{s_n + t_n\}$ is oscillating and unbounded but how can I do it for general sequences.

Answer 1

Hints: Show that either $s_n$ has a subsequence $A$ which tends to $+\infty$ as well as a subsequence $B$ which tends to $-\infty,$ or $s_n$ has a sequence $C$ going to one of $\pm \infty$ and $s_n$ has a bounded subsequence $D.$ [using oscillating assumption] Then combine each possibility with the fact that $t_n$ is bounded.

Answer 2

Hint: Bounded+ unbounded => unbounded

Unbounded => does not converge to finite

Converges to $\pm\infty$ - bounded => Converges to $\pm \infty$ , so contrapositve:

Oscilatory +bounded=> (not converge to $\pm \infty$)+ bounded => not converge to $\pm \infty$