If we let F be a non-principal ultrafilter on the Natural numbers, and define $a,b \in \mathbb R^{\mathbb N}$ to be real-valued sequences. Then the equivalence relation ~ can be defined as a~b if the set $\{j \in \mathbb N | a\_j = b\_j \} \in F$. 

I.e. that two sequences are equivalent if the set of natural numbers on which their points agree is in our ultrafilter. Now if we take the quotient of $ \mathbb R^{\mathbb N}$ / ~ then this creates the Hyperreal numbers. 

I don't understand what is happening when I take this quotient though. I'm guessing it is doing something like partitioning the real valued sequences into equivalence classes by using each of the sequencs that are inside the ultrafilter? I'm not really sure though. Any help, this step is confusing.