MHB Finding the Infimum and Supremum

[slwarrior64]

Hello, I feel like I am struggling with this more than I should. I can tell intuitively what the infimum and supremum are, but I am pretty sure that I need a more formal proof style answer. How would one actually prove this question?

 

[HOI]

One doesn't "prove" a questiom! One simply answers a question and then, prehaps, proves that the ansser is correct. You say "I can tell intuitively what the infimum and supremum are," Okay, what are they?

 

[slwarrior64]

I got that the Supremum does not exist because the set has no upper bound, and I got that the lower bound was 0 because once k reaches the negatives you are geting a negative exponent which becomes a fraction with an increasingly large denominator. I was unsure about what to do with it because the other problems we had done discussed the inf and sup of things that included inequalities so we had to rearrange the inequality and prove it true.

 

[HOI]

if you want inequalities, you know that 2^k> 0 for all k. You also can say that, given any positive integer, N, there exist a positive integer k so that 2^k>N. That certainly shows that 2^k has no upper bound. But it is also true that 2^{-k}< \frac{1}{N} showing that the infimum is 0.

 

[slwarrior64]

Okay, I think I get what you are saying, but there are some weird formatting things in your reply that I don't really understand:
if you want inequalities, you know that 2^k> 0 for all k. You also can say that, given any positive integer, N, there exist a positive integer k so that 2^k>N. That certainly shows that 2^k has no upper bound. But it is also true that 2^{-k}< \frac{1}{N} showing that the infimum is 0.