 To show that every bounded, increasing sequence has a limit, consider the Dedekind cuts defining the individual real numbers in the sequence. Let us define a new Dedekind cut by taking its left set to contain every rational in the left set of at least one of these terms of the sequence, and taking its right set to contain every rational that is in all of the right sets of these terms. Because the sequence is bounded, we know that the right set is non-empty, and the rest of the properties of a Dedekind cut are not hard to check.