Addition

As I've mentioned previously, the hard part of Real Analysis is forgetting what you know and really coming to the proofs with only what you've proven a-priori. Addition is a strong example of this because we (hopefully) all know how to add numbers. However, its not as simple as that. Lets start with a definition

Definition: Let $m$ be a natural number. To add $0$ to $m$, we define $0 + m := m$. InductivelyFrom the Peano Axioms, suppose we have defined how to add $n$ to $m$, then we can add $n$++ to $m$ by defining $(n$++$)$ $+ m := (n+m)$++. Formally, we can formulate a recursive definition to see this in action: $f_n(a_n)=a_n++$ Then, we have a series of lemmas that move us forward:

Lemma: For any natural number $n$, $n + 0 = n$
Lemma: For any natural numbers $n$ and $m$, $n + (m++) = (n + m)++$
Proposition (Commutative): For any natural numbers $n$ and $m$, $n + m = m + n$
Proposition (Associative): For any natural numbers $a,b,c$, we have $(a+b) + c = a + (b+c)$
Proposition (Cancellation Law): Let $a,b,c$ be natural numbers, if $a+b = a + c$, then $b=c$

These can all be proved via induction. Next, some ideas regarding positivity in this number system:

Definition: A natural number $n$ is said to be positive iff if it not equal to 0.
Proposition: If $a$ is a positive natural number, and $b$ is a positive natural number, then $a+b$ is a positive natural numberAnd also $b+a$ by the proposition above.
Lemma: Let $a$ be a positive number, then there exists exactly one natural $b$ such that $b++ = a$

The last stage of this section concerns itself with the ordering or natural numbers.

Definition: Let $n$ and $m$ be natural numbers. We say that $n$ is greater than or equal to $m$ iff $n = m+a$ where $a$ is any natural number.
Proposition: Let $a,b,c$ be natural numbers. Then:

1. Order is reflexive
2. Order is transitive
3. Order is antisymmetric
4. Addition preserves order
5. $a<b$ iff $a++ \leq b$
6. $a < b$ iff $b = a + d$ for some positive number $d$

Proposition (Trichotomy Law): Let $a$ and $b$ be natural numbers, then exactly one of the following statements is true:

1. $a < b$
2. $a = b$
3. $a > b$