how to create the set of hyperreal numbers using ultraproduct As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?

The explanation may be a bit simpler than seems to be implied by earlier answer. Namely, the hyperreals are a certain quotient of the space $\mathbb{R}^{\mathbb N}$ of sequences of real numbers.  More specifically, consider sequences $(x_n)$ that vanish for almost all $n$, and let $I\subset \mathbb{R}^{\mathbb N}$ be a maximal ideal containing all such sequences (the existence of a maximal ideal is proved in undergraduate algebra courses).  Then the hyperreals are a quotient field $\mathbb{R}^{\mathbb N}/I$.  This is equivalent to a certain construction using an ultrafilter on $\mathbb N$. A more detailed summary of the construction can be found on page 911 in  https://www.math.wisc.edu/~keisler/calc.html 
