Hi

You may know that, if right multiplication ( ) is strictly increasing. Therefore, the class of ordinals is non empty and has a minimum, which must be a successor (or else, let's call this minimum since right multiplication is continuous, we would have and since this means there is a such that absurd) so let's name this minimum and consider

It is such that and is maximal for this property.

Use now that right addition is also a strictly increasing continuous function, therefore there is a minimal ordinal such that you find it again to be a successor, let's say and conclude. ( Also, if since right addition is an (strictly) increasing function, you get by hypothesis on contradiction; therefore )