Learn what mathematical terms mean, Doron, and stop blaming others for your lack of knowledge. The interval [X,Y] is a finite interval, and that's a fact.
If you can't support your claim, fine, but please have the decency to cease with your empty allegations, then.
For reference, your claim was that somehow the use of the word, all, you found in a wiki article about intervals requires that any real number Y must have an immediate predecessor.
You are unable to support this allegation, so we may reject it.
I separately alleged the set { X : X < Y } for some Y has no largest element. You seem hung up on this one, too, so I will support the claim.
I will use a simple proof by contradiction. As with all such proofs, it begins with an assumption then proceeds to construct a contradiction, thereby showing the assumption to be false.
Assume the set {X : X<Y} does have a largest element, Z.
For Z to be an element of the set, Z < Y.
Let h be any element of the interval (Z,Y).
By the construction of h, Z < h < Y.
Since h < Y, h is an element of the set {X : X<Y}.
Since Z < h, the assumption Z was the largest element of the set has been contradicted.
Therefore, the set {X : X<Y} does not have a largest element.
