Stability by the Von Neumann criterion

Question

I've just determined the explicit numerical method to solve the burgers equation $$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = v \frac{\partial^2 u}{\partial x^2}$$ in one dimension with the upwind strategy for convective terms. $$u_{i,j+1} = -\delta t * CONV(u) + r*v*u_{i+1,j}+(1-2*r*v) u_{i,j}+r*v*u_{i-1,j}$$ where $r=\frac{\delta t}{\delta x^2}$, $v=\frac{1}{Re}$ where $Re$ is the Reynolds' number, and $$CONV(u)=\frac{\bar{u}_{i+1/2,j}* u_{i+1/2,j} - \bar{u}_{i-1/2,j}*u_{i-1/2,j}}{2\delta x}$$ In which the variables $$\bar{u}_{i+1/2,j}=\frac{1}{2}*(u_{i+1,j}+u_{i,j})$$ $$\bar{u}_{i-1/2,j}=\frac{1}{2}*(u_{i,j}+u_{i-1,j})$$ So if $$ \bar{u}_{i+1/2,j} > 0 \rightarrow u_{i+1/2,j}=u_{i,j}$$ $$ \bar{u}_{i+1/2,j} \leq 0 \rightarrow u_{i+1/2,j}=u_{i+1,j}$$ $$ \bar{u}_{i-1/2,j} > 0 \rightarrow u_{i-1/2,j}=u_{i-1,j}$$ $$ \bar{u}_{i-1/2,j} \leq 0 \rightarrow u_{i-1/2,j}=u_{i,j}$$ Now, I want to evaluate the stability of the numerical method I have just determined using the Von Neumann criterion.To do so, do the following

1. Replace $u_{i+m}^{n+k} = V^{n+k}e^{I(i+m)\phi}$, where $I=\sqrt{-1}$ is the imaginary number and $C=rv$ $$V^{n+1} e^{I (i) \phi} = -\delta t * CONV(u) + C * V^{n} * e^{I(i+1)\phi} +(1-2C) *V^n *e^{I(i)\phi} + C*V^n*e^{I(i-1)\phi} $$
2. Simplify by $e^{I(i)\phi}$ on both sides of equality $$V^{n+1} = -\frac{\delta t * CONV(u)}{e^{I(i)\phi}} + C * V^{n} * e^{I\phi} +(1-2C) V^n + C*V^n*e^{-I\phi} $$
3. We take out common factor $$V^{n+1} = -\frac{\delta t * CONV(u)}{e^{I(i)\phi}} + V^{n}(C(e^{I\phi}+e^{-I\phi}-2)+1) $$
4. Replace $e^{I\phi}+e^{-I\phi}=2\cos(\phi)$ $$V^{n+1} = -\frac{\delta t * CONV(u)}{e^{I(i)\phi}} + V^{n}(C(2\cos(\phi)-2)+1) $$
5. We solve $CONV(u)$ and replace $u_{i+m}^{n+k} = V^{n+k}e^{I(i+m)\phi}$ $$CONV(u) = \frac{\bar{u}_{i+1/2,j}* u_{i+1/2,j} - \bar{u}_{i-1/2,j}*u_{i-1/2,j}}{2\delta x}$$ $$CONV(u) = \frac{(\frac{1}{2}(u_{i+1,j}+u_{i,j}))* u_{i+1/2,j} - (\frac{1}{2}(u_{i,j}+u_{i-1,j}))*u_{i-1/2,j}}{2\delta x} $$ $$CONV(u) = \frac{(u_{i+1,j}+u_{i,j})*u_{i+1/2,j} - (u_{i,j}+u_{i-1,j})*u_{i-1/2,j}}{4\delta x}$$ $$CONV(u) = \frac{(V^n*e^{I(i+1)\phi}+V^n*e^{I(i)\phi})*V^n*e^{I(i+1/2)\phi} - (V^n*e^{I(i)\phi}+V^n*e^{I(i-1)\phi})*V^n*e^{I(i-1/2)\phi}}{4\delta x}$$ $$CONV(u) = \frac{ V^{2n} * e^{I (i+1/2) \phi} (e^{I(i+1)\phi}+e^{I(i)\phi}) - V^{2n} * e^{I(i-1/2)\phi}(e^{I(i)\phi} + e^{I(i-1)\phi}) }{4\delta x}$$
6. Solve $\frac{CONV(u)}{e^{I(i)\phi}}$ as in step 4 $$\frac{CONV(u)}{e^{I(i)\phi}} = \frac{ V^{2n} * e^{(I\phi)/2} (e^{I\phi}+1) - V^{2n} * e^{-(I\phi)/2}(e^{-I\phi} + 1) }{4\delta x}$$ $$\frac{CONV(u)}{e^{I(i)\phi}} = \frac{ 2V^{2n} * e^{I\phi}*\cos(\frac{\phi}{2}) - 2V^{2n} * e^{-I\phi}\cos(\frac{\phi}{2}) }{4\delta x}$$ $$\frac{CONV(u)}{e^{I(i)\phi}} = \frac{ 2V^{2n} (e^{I\phi}*\cos(\frac{\phi}{2}) - e^{-I\phi}\cos(\frac{\phi}{2})) }{4\delta x}$$ $$\frac{CONV(u)}{e^{I(i)\phi}} = \frac{ V^{2n} (I\sin^2(\phi) \csc(\frac{\phi}{2})) }{2\delta x}$$
7. Now replace step 6 in step 4 $$V^{n+1} = -\frac{\delta t * V^{2n} (I\sin^2(\phi) \csc(\frac{\phi}{2}))}{2\delta x} + V^{n}(C(2\cos(\phi)-2)+1) $$

That's what I've, let's remember that to prove stability with the von neumann criterion we have to prove that $|\frac{V^{n+1}}{V^n}| \leq 1$ but I don't know how to clear $V^{n}$ from there, if someone could help me or explain or tell me if I'm wrong, I would appreciate it.