The Sum of $n$ Ones is Strictly Increasing in $n$

Let $\mathbb{N}$ be the set of natural numbers, with successor map $S$ and order relations $<$ and $\le$, and for $n\in\mathbb{N}$ let $[n]$ be the initial segment it determines. Let $\mathbb{R}$ be the ordered field of real numbers, with additive identity $0$ and multiplicative identity $1$.

For $n\in\mathbb{N}$ let $\mathbf{1}^{(n)}\in\mathbb{R}^{n}$ be the $n$-tuple with every component equal to $1$, and let $\sigma:\mathbb{N}\to\mathbb{R}$ be given by the finite sum

$$\sigma(n)=\sum_{k=1}^{n}\mathbf{1}^{(n)}_{k}.$$

Then the following hold for all $m,n\in\mathbb{N}$, inequalities between natural numbers being those of $\mathbb{N}$ and inequalities between values of $\sigma$ those of $\mathbb{R}$.

1. (Recursion) $\sigma(1)=1$ and $\sigma(S(n))=\sigma(n)+1$.

2. (Strict monotonicity) If $m<n$, then $\sigma(m)+1\le\sigma(n)$; in particular $\sigma(m)\le\sigma(n)$ and $\sigma(m)\ne\sigma(n)$.

3. (Injectivity) If $\sigma(m)=\sigma(n)$, then $m=n$.