Show that for each infinite cardinal , .

Notation: denotes the cofinality. I know some properties of . They may be helpful.

For each infinite cardinal number , .

However, I do not see how to prove this. Any hints would be great. Thanks.

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- March 24th 2010, 10:22 AMxianghu21cofinality, infinite cardinal
Show that for each infinite cardinal , .

Notation: denotes the cofinality. I know some properties of . They may be helpful.

For each infinite cardinal number , .

However, I do not see how to prove this. Any hints would be great. Thanks. - March 25th 2010, 06:47 AMclic-clac
Here one of Koenig's lemmas helps:

Consider two families of cardinals indexed by such that for any then:

Use this result with the fact that given a cardinal

is the lowest cardinal such that there exists a family with for all and