Real Analysis, 131A, Fall 2014 (full set of notes)

... Lemma 1.2.7. For any natural number n, n + 0 = n. Remark 1.2.8. Note that we cannot apply commutativity of addition, since it does not immediately follow from the axioms of Definition 1.2.1. Proof. From Definition 1.2.3, 0 + 0 = 0. So, we induct on n. Suppose n + 0 = n for a natural number n. We nee ...

... Lemma 1.2.7. For any natural number n, n + 0 = n. Remark 1.2.8. Note that we cannot apply commutativity of addition, since it does not immediately follow from the axioms of Definition 1.2.1. Proof. From Definition 1.2.3, 0 + 0 = 0. So, we induct on n. Suppose n + 0 = n for a natural number n. We nee ...

CONSTRUCTION OF THE REAL NUMBERS. 1. Motivation. It will not

... that, from a formal point of view, the question is meaningless: For since we have not yet constructed the real numbers we can not attach any precise meaning to the phrase ‘approximating real numbers’ ... But that the question is formally meaningless does not mean that it’s mathematically pointless o ...

... that, from a formal point of view, the question is meaningless: For since we have not yet constructed the real numbers we can not attach any precise meaning to the phrase ‘approximating real numbers’ ... But that the question is formally meaningless does not mean that it’s mathematically pointless o ...