Hi i was wondering if anyone could help me show the following
$f\le^{*}g \iff \exists k\forall m\ge k(f(m)\le g(m)$ Show that $\le^{*}$ is a lattice.
My definition of a lattice is the following let $\le$ be a partial ordering on $X$ and that $\sup\{x,y\}$ and $\inf\{x,y\}$ exist in $X$ $\forall x,y\in X$ then $\le$ is a lattice.?
I'm unsure where to start this problem, i know that i need to show that the $\inf\{x,y\}$ and $\sup\{x,y\}$ exists but how would i do that?
As was pointed out, <=* is not a porder.
It is only a preorder. To rescue the problem for
functions f:R -> R a porder instead of a preorder is needed.
Define f eventually = g, f == g as
some k in R with for all x >= k, f(x) = g(x)
Show == is an equivalent relation.
To show the equivalent classes are a lattice
using f eventually <= g, f <=* g when
some k in R with for all x >= k, f(x) <= g(x).
Define [f] <<= [g] when f <=* g.
Show that <<= is well defined.
That is, if f ~ h, g ~ k, f <=* g, then h <=* k.
Continue to show <<= is a partial order.
Finally show [max(f,g)] = sup([f],[g]) and define inf.
Whence this curve ball problem?
Did you include all the details?