I am extremely new to proofs, and quite bad at them. In studying and practicing the different types of proofs, I developed this very rough proof that $1+1=2$, one of the simplest mathematical truths I could think of, but due to my ignorance I cannot tell if it is valid. Is the following a valid proof, and if not, how can it be improved upon?
Proof:
Let $a=1$. It is true that $a \in \mathbb N \setminus \{0\}$ and $\not\exists x \in \mathbb N\setminus \{0\}$ such that $x < a$. Therefore, $a$ is the least element of $\mathbb N \setminus \{0\}$.
Let $m$ be the least element of $\mathbb N \setminus \{0,1\}$. $2$ is the least element of $\mathbb N \setminus \{0,1\}$, therefore $m=2$.
Assume that $1+1 \neq 2$. There are two possible cases:
- Case 1: $2<1+1$. If this is true, then because a set of natural numbers only contains positive whole numbers, $2=1$. Because $2$ is an element of $\mathbb N \setminus \{0,1\}$, $2 \neq 1$. Therefore this case leads to a contradiction.
- Case 2: $2>1+1$. If this is true, then there must exist a natural number $z < 2$ such that $z=1+1$. Because two is the least element of $\mathbb N \setminus \{0,1\}$, $2=z$. However, this contradicts the statement that $z<2$, and therefore this case leads to a contradiction.
Because assuming that $1+1 \neq 2$ leads to contradiction, it must be true that $1+1=2$.
