help to prove

**dbhakta** · April 23rd 2008, 08:52 AM

prove if a A is uncountable and A is a subset of B, then B is uncountable

**ThePerfectHacker** · April 23rd 2008, 10:03 AM

Originally Posted by dbhakta

prove if a A is uncountable and A is a subset of B, then B is uncountable

Since $A$ is uncountable it means $|A| > \aleph_0$ . Since $A\subseteq B$ it means $|A|\leq |B|$ because the identity map on $A$ will be an injective function. Thus, $\aleph_0 < |A| \leq |B|$ .

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