Question

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

Answer 1

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.

Answer 2

ok the dbl lines ontop refer to cardinality

Answer 3

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