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Please help me with the question:

  1. Let $X \subseteq \mathbb{R}$ such that $\sup X$ exists . Which subsets of $X$ have a supremum?

  2. Let $Y$ be a subset of $X$ such that $Y$ is not empty. What is the the relationship between $\sup Y$ and $\sup X$?

1 Answers 1

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Recall that completeness of $\mathbb{R}$ states that every non-empty subset of $\mathbb{R}$ which is bounded above has a supremum.

If $X \subseteq \mathbb{R}$ has a supremum and $Y \subseteq X$ then $Y$ is bounded above (why?) and so if it is non-empty it also has a supremum. If $y \in Y$ then what do we know about $y$ in relation to $\sup X$ (can we say something about which one's bigger? Recall the definition of supremum).

Using the definition of supremum as the least upperbound, what does this tell us about $\sup Y$?

Edit: I believe this is what you were asking but could you clarify what you mean by

$X \subseteq \mathbb{R}$ such that $\sup X$

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    I think it is : such that $\sup X<+\infty$. – 2017-02-27
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    Such that supremum of X exists. – 2017-02-27
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    Thanks. Anyone to explain this for me or to give me an answer. I am trying it but I am stacked. I just started doing Real analysis this month. – 2017-02-27
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    Have you tried answering the questions in my answer? If so let me know where you're stuck – 2017-02-28