0
$\begingroup$

Please help me with the question:

Let $X\subseteq \mathbb{R}$ such that $\sup X$ exists . Which subsets of $X$ have supremum? Let $Y$ be a subset of $X$ such that $Y$ is not empty. What is the the relationship between $\sup Y$ and $\sup X$.

  • 2
    In general nothing can be said about the existence of a supremum of $Y$. But if it exists, $\sup Y \leq \sup X$. – 2017-02-28
  • 0
    possibly duplicated http://math.stackexchange.com/questions/909111/supremum-of-a-subset-is-less-or-equal-than-infimum-of-another-subset?rq=1 – 2017-02-28
  • 0
    @RafaelWagner The other question referred to is not a duplicate of this one, because its premise $\forall x \in X, \forall y \in Y : x \leq y$ is not equivalent to one set being a subset of the other. – 2017-02-28
  • 0
    @SimonMarynissen I'll probably kick myself as soon as I've posted this, but surely something *can* be said generally about the supremum of $Y$, given the stated premise that $Y$ is not empty? I'll get me coat. – 2017-02-28
  • 0
    $Y$ is bounded above, and @CalumGilhooley ok – 2017-02-28

0 Answers 0