Is this equivalence relation correct?

Question

The norms $\def\n#1{\lVert\,\rVert_{#1}} \n1$ and $\n2$ are said equivalent, if and only if, there are real numbers, $\alpha > 0, \beta > 0$ such that $\alpha \n1 \leq \n2 \leq \beta \n1$.

Consider the relation $\n1 \sim \n2$ if and only if $\n1$ and $\n2$ are equivalent, check that $\sim$ is an equivalence relation.

Solución: where $\alpha, \beta > 0$ if:

$\alpha \n1 \leq \n2$ then $\n1 \leq \frac{1}{\alpha} \n2$

$\n2 \leq \beta \n1$ then $\frac{1}{\beta} \n2 \leq \n1$

For the transitive, if:

$\alpha\, \n1 \leq \n2 \leq \beta\, \n1$

$\alpha' \n2 \leq \n3 \leq \beta' \n2$

Then:

$\alpha\alpha'\n1 \leq \n3 \leq \beta\beta'\n1$

this right? If not, I appreciate you can help me with a better way to write it.

Answer 1

You have all the important parts of the argument for the symmetric and transitive properties. But it would be nice to show conclusions explicitly in each case. Since the relation is defined by the existence of these positive real scalars, it's nice to explicitly name the scalars in the conclusion.

For instance, for symmetric property: $$ \lVert\,\rVert_1 \sim \lVert\,\rVert_2 \quad\Rightarrow\quad \alpha \lVert\,\rVert_1 \leq \lVert\,\rVert_2 \leq \beta \lVert\,\rVert_1 \quad\Rightarrow\quad \tfrac{1}{\beta} \lVert\,\rVert_2 \leq \lVert\,\rVert_1 \leq \tfrac{1}{\alpha} \lVert\,\rVert_2 \quad\Rightarrow\quad \lVert\,\rVert_2 \sim \lVert\,\rVert_1 $$

For the transitive property: $$ \left. \begin{aligned} \lVert\,\rVert_1 \sim \lVert\,\rVert_2 \;&\Rightarrow\; \,\alpha\, \lVert\,\rVert_1 \leq \lVert\,\rVert_2 \leq \beta\, \lVert\,\rVert_1 \\ \lVert\,\rVert_2 \sim \lVert\,\rVert_3 \;&\Rightarrow\; \alpha' \lVert\,\rVert_2 \leq \lVert\,\rVert_3 \leq \beta' \lVert\,\rVert_2 \end{aligned} \;\right\}\!\!\!\Rightarrow \; \alpha^{}\alpha' \lVert\,\rVert_1 \leq \lVert\,\rVert_3 \leq \beta^{}\beta' \lVert\,\rVert_1 \;\Rightarrow\; \lVert\,\rVert_1 \sim \lVert\,\rVert_3 $$

For the reflexive property, just note that both scalars can be $1$: $$ 1\, \lVert\,\rVert_1 \leq \lVert\,\rVert_1 \leq 1\, \lVert\,\rVert_1 \quad\Rightarrow\quad \lVert\,\rVert_1 \sim \lVert\,\rVert_1 $$