Is a chain complete lattice complete? If every chain in a lattice is complete (we take the empty set to be a chain), does that mean that the lattice is complete? If yes, why? 
My intuition says yes, and the reasoning is that we should somehow be able to define a supremum of any subset of the lattice to be the same as the supremum of some chain related to that lattice, but I've not abled to make more progress on the same. ANy suggestions? 

If $L$ is not complete, it has a subset with no join; among such subsets let $A$ be one of minimal cardinality, say $A=\{a_\xi:\xi<\kappa\}$ for some cardinal $\kappa$. For each $\eta<\kappa$ let $A_\eta=\{a_\xi:\xi\le\eta\}$; $|A_\eta|<\kappa$, so $A_\eta$ has a join $b_\eta$. Clearly $b_\xi\le b_\eta$ whenever $\xi\le\eta<\kappa$, so $\{b_\xi:\xi<\kappa\}$ is a chain; indeed, with a little more argumentation we can assume that the chain is a strictly increasing $\kappa$-sequence. Now let $b=\bigvee_{\xi<\kappa}b_\xi$ and show that $b=\bigvee A$ to get a contradiction.
