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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.

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

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