A well ordered set is a total ordered set in which every finite subset has a least element.

Now (Q ∩ [0, 1], ≤) will eventually give ([0,1],<=) wouldn't it ?

Now [0,1] basically contains a subset (0,1) which doesn't have least element , because here if you choose any number then i can always give you a smaller number than that.

In total ordered set we look for the whole set as well as all the subsets so cant i say (0,1) is a subset of [0,1] for which there is no least element ?

Its important to see for whole set and every finite subset of it :