Let $(E,\leq)$ a partially ordered set. Zorn lemma says that if all chain of $E$ has a supremum, then $E$ has a maximal element.
So if I consider, $\Big((0,1),\leq \Big)$, it has no maximal element but all chain is upper bounded by $1$, so it doesn't work here, no ?
Your partial order $((0,1),{\le})$ is a total order, so $(0,1)$ is itself a chain. It has no upper bound! You propose that $1$ is an upper bound, but $1$ isn't even a member of the partial order you're considering, so that doesn't count.
Note, by the way, that Zorn's Lemma doesn't demand that every chain must have a supremum (which means a least upper bound) -- it is enough that every chain has some upper bound in the partial order.