Question: [Banach limit](http://en.wikipedia.org/wiki/Banach_limit), as a non-constructive object, is not unique. The Banach limit for some sequences, say, convergent sequences, sequences satisfying $a\_n = a\_{n+m}$ for all $n$ and some $m$, the Banach limit is uniquely determined, but there also exists some bounded sequence $\{a\_n\}\_{n \in \mathbb N}$ whose Banach limit is not unique. Can we formally construct such a sequence, say, $a\_n = f(a\_{n-1})$, with a explict function $f$, but its Banach limit is not unique for all Banach limit?

Answer: 
As Andreas Caranti pointed out, there do exist finite extensions that are splitting fields and yet are not separable. I think now that the best way to make sense of the terminology is to abandon the idea of a "finite normal extension." Leave the definition of a **normal extension** $E$ of $F$ as a splitting field over $F$, then based on [this](http://en.wikipedia.org/wiki/Galois_extension), define a **Galois extension** $E$ of $F$ as an extension that is both normal and separable. Then, we can just write "finite Galois extension" wherever Fraleigh wrote "finite normal extension."
No, a finite extension can be a splitting field, and still not be separable.

The standard example consists in taking a function field $E = F(t)$ over the field $F$ with $p$ elements, $p$ a prime. Then one takes the subfield $K = F(t^{p})$. Now $E$ is the splitting field over $K$ of the polynomial $f = x^{p} - t^{p} \in K[x]$, which can be shown to be irreducible in $K[x]$ and $E/K$ is not separable, as $f = (x - t)^{p}$ has multiple roots.