OpenStudy (swissgirl):

Is the following proof correct? Every subset of a countable set is countable:

5 years ago
OpenStudy (swissgirl):

Proof: Let A be a countable set and let $$B \subseteq A$$. If B is finite then B is countable by definition, If B is infinite since $$B \subseteq A$$, A is infinite. Thus A is denumerable. By Theorem 5.5.4, B has a denumerable subset C. Thus $$C \subseteq B \subseteq A$$ which implies $$\aleph_0 = \overline { \overline {C}}$$ and $$\overline { \overline {C}} \leq \overline { \overline {B}} \leq \overline { \overline {A}} = \aleph_0$$. Therefore $$\overline { \overline {A}}=\overline { \overline {B}}= \aleph_0$$. Thus B is denumerable and hence countable.

5 years ago
OpenStudy (swissgirl):

ok the dbl lines ontop refer to cardinality

5 years ago
OpenStudy (swissgirl):

There usually can be one sill mistake but like I can never catch them

5 years ago