Definition of Banach limit In my Bachelor Thesis I have defined a Banach limit as a functional $\operatorname{LIM} : l^\infty (\mathbb{N})\rightarrow \mathbb R$ that has the following properties:
B1 If $(x_n)$ is a convergent sequence, then $\operatorname{LIM}(x_n)=\lim_{n\to \infty} x_n$.
B2 If $x_n\geq 0$ for all $n\in\mathbb N$, then $\operatorname{LIM}(x_n)\geq 0$.
B3 $\forall\alpha,\beta\in\mathbb{R}\forall (x_n),(y_n)\in l^\infty(\mathbb{N})[\operatorname{LIM}(\alpha (x_n)+\beta (y_n)) = \alpha \operatorname{LIM}((x_n)) + \beta \operatorname{LIM}((y_n))]$.
B4 If $y_n=x_{n+1}$ for all $n\in\mathbb{N}$, then $\operatorname{LIM}((x_n))=\operatorname{LIM}((y_n))$.
My supervisor thought that the condition B2 might not be necessary, because maybe it followed from the other conditions. I've looked up a lot of definitions of Banach limits, but they're all a little different, which makes it hard to compare sometimes. But it looks like most of the time, the conditions are at least as strong as mine.
I have not been able to prove that B2 is necessary, nor have I been able to find a counterexample. Can anybody tell me if the other three conditions are sufficient? If not, can you give me a counterexample?
I figured that maybe I haven't been able to find a counterexample, because there are only non-constructive counterexamples.
 A: Suppose $L_1$ and $L_2$ are two Banach limits that do not agree on some positive sequence $\vec{a} = \langle a_n : n \geq 1\rangle \in l^{\infty}$. In fact, by scaling the example given below, we can assume that $L_1(\vec{a}) = 1 $ and $L_2(\vec{a}) = 3$. Then $2L_1 - L_2$ is a counterexample.
To construct such $L_1, L_2$, first get a sequence $\vec{a} = \langle a_n : n \geq 1\rangle$ such that $a_n > 0$, $|a_{n+1} - a_n| < 1/n$ and $\{a_n : n \geq 1\}$ is the set of all positive rationals in $[0, 1]$. Then use Hahn Banach theorem to show that for every $0 < x < 1$, there is a Banach limit sending $\vec{a}$ to $x$.
A: I recommend against B2 for logic design reasons.  
Since you are studying series and limits, you probably have seen this series, 1 + 2 + 3 + 4 ....  Depending on your definition of infinite sum, it is consistent to define 
$$\sum_{k = 1}^\infty k = -\frac 1{12}$$
as Ramanujan argued to G H Hardy.
We could define $(\forall k) X_k \ge 0 \Rightarrow (\sum_{k=1}^\infty X_k) \ge 0$, and accept that divergent series have no sum.  We could define $a + b \ge a$ and accept that negative numbers have no sum.  We could define B2 and accept that LIM doesn't exist for divergent sequences, but there isn't really much to gain from that.  You can just as well have a theorem $\text{convergent}(x) \Rightarrow B2$, and have one less axiom in your definitions and much more flexibility.
