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**tn11631** (b) Show that for every $\displaystyle \alpha\in S_\Omega$, the subset $\displaystyle \{x|\alpha<x\}$ is uncountable.

Can I say: Let $\displaystyle S=\{x|\alpha<x\}$ and let $\displaystyle T=\{x|x\leqslant\alpha\}$ be a countable set. Then $\displaystyle S_\Omega = S\cup T$. Since $\displaystyle S_\Omega$ is uncountable (defined in he beg. of the problem), then either S or T has to be uncountable and seeing as T is already defined as a countable set it follows that S must not be countable. Thus, the set $\displaystyle \{x|\alpha<x\}$ is uncountable.