Example 4

One has $x^2 + xy + y^2 \simeq x^2 + y^2$ when $(x,\ y)$ tends to $(0,\ 0)$ in $\mathbf{R}^2$, but not $xy \simeq x^2 + y^2$.

It follows immediately from prop. 3 of V, p. 213, that if $f_1,\ f_2,\ g_1,\ g_2$ are functions in $\mathscr{H}(\mathfrak{F}, K)$ ($K$ being any valued field) the relations $f_1 \simeq g_1$ and $f_2 \simeq g_2$ imply that $f_1 f_2 \simeq g_1 g_2$.

We remark that in contrast the relations $f_1 \simeq g_1$ and $f_2 \simeq g_2$ do not imply that $f_1 + f_2 \simeq g_1 + g_2$, as is shown by the example $f_1(x) = g_1(x) = x^2,\ f_2(x) = -(x^2 + x),\ g_2(x) = -(x^2 - 1)$, as the real variable $x$ tends to $+\infty$.

The comparison relations $f \preccurlyeq g,\ f \simeq g$ are said to be weak. We say that two functions $f,\ g$ from $\mathscr{H}(\mathfrak{F}, V)$ are weakly comparable if they satisfy one (at least) of the relations $f \preccurlyeq g,\ g \preccurlyeq f$.