Is the limit point of a chain necessarily a least upper bound?

Question

Let the space $X,d$ have a partial order $\preceq$

Let every infinite chain $x_i\preceq x_{i+1},\ldots$ have a limit point in $X,d$

If every limit point is a least upper bound, then Zorn's Lemma applies and there is at least one maximal element.

But does it follow that every limit point of a partial order is a least upper bound?

Answer 1

Consider the following order on the real numbers from $0$ to $1$. The restriction on the set or rational (irrational) numbers is the usual order but every irrational number is smaller than every rational number. Then every infinite sequence has a limit point but the limit point of a chain can be smaller than every element of the chain.