0

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 $\sup X$.

Simon Marynissen
  • 1,357
Tomas
  • 3
  • 2
    In general nothing can be said about the existence of a supremum of $Y$. But if it exists, $\sup Y \leq \sup X$. – Simon Marynissen Feb 28 '17 at 00:13
  • possibly duplicated http://math.stackexchange.com/questions/909111/supremum-of-a-subset-is-less-or-equal-than-infimum-of-another-subset?rq=1 – R.W Feb 28 '17 at 00:26
  • @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. – Calum Gilhooley Feb 28 '17 at 01:36
  • @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. – Calum Gilhooley Feb 28 '17 at 01:38
  • $Y$ is bounded above, and @CalumGilhooley ok – R.W Feb 28 '17 at 01:40

0 Answers0