Analysis- Supremum and infimum

Question

I tried to do this by taking $ X=\{1,3,5,7\}$ and $Y =$ set of all odd natural numbers.

In this case the inf$(A)$ is negative infinity. And sup$(A)$ is finite.

But is it enough to answer the question? I doubt it is not.

Answer 1

No, it is not enough to provide an example. The idea is to determine which of the optins always takes place. And, yes, option (B) is the correct one, because:

• If $k$ is the greatest element of $X$, then every element of $A$ is smaller than $K$ and so $\sup(A)<\infty$;
• Take $k\in X$. Than $A$ contains every integer of the form $k-m$, with $m\in Y$. An infinite set of integers with an upper bound has no lower bound, and so $\inf(A)=-\infty$.

Answer 2

Since the infimum is $-\infty$, while the supremum is finite, that is, $\sup A < +\infty$, the correct answer is B. You can show that this holds in general, as in the exercise, not only in your particular case.