Let $\kappa$ be a regular cardinal and let $\langle\hat{Q}_\beta\rangle_{\beta<\alpha}$ be a finite support iterated forcing where for every $\beta<\alpha$ , $\Vdash_{P_\beta} \hat{Q}_\beta$ has the $\kappa$ chain condition.

By induction:

$P_0$ is the empty set.

If $P_\alpha$ satisfies the $\kappa$ chain condition then so does $P_{\alpha+1}$ , since $P_{\alpha+1}$ is equivalent to $P_\alpha*Q_\alpha$ and composition preserves the $\kappa$ chain condition for regular $\kappa$ .

Suppose $\alpha$ is a limit ordinal and $P_\beta$ satisfies the $\kappa$ chain condition for all $\beta<\alpha$ . Let $S=\langle p_i\rangle_{i<\kappa}$ be a subset of $P_{\alpha}$ of size $\kappa$ . The domains of the elements of $p_i$ form $\kappa$ finite subsets of $\alpha$ , so if $\operatorname{cf}(\alpha)>\kappa$ then these are bounded, and by the inductive hypothesis, two of them are compatible.

Otherwise, if $\operatorname{cf}(\alpha)<\kappa$ , let $\langle \alpha_j\rangle_{j<\operatorname{cf}(\alpha)}$ be an increasing sequence of ordinals cofinal in $\alpha$ . Then for any $i<\kappa$ there is some $n(i)<\operatorname{cf}(\alpha)$ such that $\operatorname{dom}(p_i)\subseteq \alpha_{n(i)}$ . Since $\kappa$ is regular and this is a partition of $\kappa$ into fewer than $\kappa$ pieces, one piece must have size $\kappa$ , that is, there is some $j$ such that $j=n(i)$ for $\kappa$ values of $i$ , and so $\{p_i\mid n(i)=j\}$ is a set of conditions of size $\kappa$ contained in $P_{\alpha_j}$ , and therefore contains compatible members by the induction hypothesis.

Finally, if $\operatorname{cf}(\alpha)=\kappa$ , let $C=\langle \alpha_j\rangle_{j<\kappa}$ be a strictly increasing, continuous sequence cofinal in $\alpha$ . Then for every $i<\kappa$ there is some $n(i)<\kappa$ such that $\operatorname{dom}(p_i)\subseteq \alpha_{n(i)}$ . When $n(i)$ is a limit ordinal, since $C$ is continuous, there is also (since $\operatorname{dom}(p_i)$ is finite) some $f(i)<i$ such that $\operatorname{dom}(p_i)\cap [\alpha_{f(i)},\alpha_i)=\emptyset$ . Consider the set $E$ of elements $i$ such that $i$ is a limit ordinal and for any $j<i$ , $n(j)<i$ . This is a club, so by Fodor's lemma there is some $j$ such that $\{i\mid f(i)=j\}$ is stationary.

For each $p_i$ such that $f(i)=j$ , consider $p^\prime_i=p_i\upharpoonright j$ . There are $\kappa$ of these, all members of $P_j$ , so two of them must be compatible, and hence those two are also compatible in $P$ .