My teacher has not gone over the monotonicity theorem, but I suspect that this would be of great help. Could someone please help me understand if I can apply this to the question or correct me if I am mistaken? Thank you.

  • $\begingroup$ Is the left hand side supposed to be the supremum of $X \cup Y$? $\endgroup$ – Alan Jun 14 '15 at 6:19

You can just do this by the definitions. Let $\alpha = \sup\Big({\sup{X},\sup{Y}}\Big)$ and let $\beta = \sup(X\cup Y)$.

First, we show $\alpha \leq \beta$. There's a sequence in $X\cup Y$ converging to $\beta$. This sequence has infinitely many elements in either $X$ or $Y$. Without loss, assume it's $X$. Then $\sup X \leq \beta$. And $\alpha \leq \sup X $. So $\alpha \leq \beta$.

To show $\beta \leq \alpha$, any sequence in $X$ converging to $\sup X$ also a sequence in $X\cup Y$. So $\beta \leq \sup X$. Similarly, $\beta \leq \sup Y$. So $\beta \leq \alpha$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.