True or False? Sequence Question for {sn+tn}

[happyg1]

Hi,
Here's the question:
If {sn} is oscillating and not bounded, and {tn} is bounded, then {sn+tn} is oscillating and not bounded.
True or False?

I have tried for a while to find a counterexample, but I can't. I am leaning towards saying that this is true...but then I have to prove it. Am I correct in saying that it's true?
I have also started trying to prove it, but I'm having a hard time. Any clarification or advice will be appreciated.
CC

 

[AKG]

{sn} is not bounded, and let's say that it's not bounded above in particular. Let t be inf{tn}. Then (sn + tn) > (sn + t) for all n, so if {sn + t} is not bounded above, then neither is {sn + tn}. But if {sn + t} is bounded, then there is some K such that sn + t < K for all n, so sn < K - t for all n, so {sn} is bounded above, contradiction.

 

[happyg1]

Hi,
Thanks for the help. I was trying to prove it directly and I wound up with a bunch of nonsense. I understand the problem now, and I have worked the other 3 that were giving me headaches.
Thanks
CC

 

[HallsofIvy]

What, precisely, do you mean by "oscillating"? I have seen it used to mean that terms alternate sign or simply to mean that s_n-1< s_n but s_n> s_n+1.

 

[happyg1]

Our definition of oscillating is a sequence that diverges but not to + infinity or - infinity.