In the ultrafilter construction of the hypperreals, is ordering done component wise? I.e. if $a,b$ are hyperreal numbers then $a = [x_n]$ and $ b = [y_n]$ where $[x_n]$ and $[y_n]$ are equivalence classes of the set $\mathbb{R}^\mathbb{N}$ under the equivalence realtion: $(r_n) \sim (s_n)$ if $\{n: r_n = s_n\}\in \mathcal{F}$ where $\mathcal{F}$ is a non-principle ultrafilter on $\mathbb{N}$.
Then do we say $[x_n]<[y_n]$ if $\{n: x_n<y_n\}\in \mathcal{F}$?