Equivalence between "cofinality" definitions

Question

I am using the very standard definition of cofinality, which is:

For the limit ordinal $\lambda$, we define its cofinality as:

$cof(\lambda):=$ min { $card(x)$ | $ x\subset \lambda$ is cofinal in $\lambda$ }

Though, recently I've read in some books different version, were instead of $card(x)$ they use $otp(x)$ with everything else the same. Here, $otp(x)$ stands for ordertype.

Now I want to show that these definitions are equivalent.

One inequality is obvious since $otp(x) \geq card(x)$ and hence taking minimums doesn't change anything, but how can I prove the opposite sign...

Answer 1

Suppose you have a cofinal $x\subseteq \lambda$ with $|x|=\alpha$.

Then, by definition you can well-order $x$ with order type $\alpha$.

Consider the subset $y$ of elements of $x$ that are larger (according to the natural order) than every earlier (according to the new well-ordering) elements of $x$. Then $y$ is still cofinal in $\lambda$: By induction on the new well-ordering, every element of $x$ has some element of $y$ as an upper bound.

On the other hand, by construcion, on $y$ the new well-order agrees with the natural order -- so the order type of $y$ is at most $\alpha$.