How to prove that the length function of a half-open interval is monotone?

Question

Let $\mathcal{I} = \{ [a,b) : a \leq b, a,b \in \mathbb{R} \}$

Prove that if $I \subseteq J \implies \mathcal{l}(I) \leq \mathcal{l}(J)$ where $I,J \in \mathcal{I}$

The proof seems trivial and intuitive but, I want to write it formally:

Here is my attempt:

Let $I \subseteq J$ where $I,J \in \mathcal{I}$

Define: $I= \{x \in \mathbb{R} | b \leq x \leq c \}$

and $J= \{y \in \mathbb{R} | a \leq y \leq d \}$

So we have $a \leq b \leq c \leq d$

$\mathcal{l}(I)= c-b$

$\mathcal{l}(J)= d-a$

But since $c \leq d$ and $a \leq b \implies -a \geq -b$

Thus, $c-b \leq d-a \implies \mathcal{l}(I) \leq \mathcal{l}(J)$

Answer 1

First off, note you've written $I = [b,c]$ and $J=[a,d]$ rather than the intended $I = [b,c)$ and $J=[a,d)$. It's entirely possible this is just a typo.

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Also, if you want to write it formally, you need to start with $I \subseteq J$ and $I,J \in \mathcal{I}$, not $a \leq b \leq c \leq d$ and $I = [b,c), J = [a,d).$

That is, you'll need to assume $a \leq d$ and $b \leq c$ where $[b,c) \subseteq [a,d)$ and prove $a\leq b$ (almost immediate) and $c\leq d$ (requires a short argument).

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Other than those two issues, it's fine.