real analysis – Limit of a system of equations

I have the following system of equations:
begin{align}
frac{d}{dt}u_n(t)=u_n(t)Big(u_{n+1}(t)-u_{n-1}(t)Big), qquad nin {1, cdots, N}.
end{align}

I am reading a paper in which they say that in the space-continuous limit this system of equations turns into the system
begin{align}
frac{partial}{partial t}u(x, t)+u(x,t)frac{partial}{partial x}u(x, t)=0.
end{align}

In which sense is that true?

I was thinking the following:

I consider $n$ as a discrete spacial variable and I scale the discrete system as follows:
begin{align}
frac{d}{dt}u_{frac{n}{N}}(t)=u_{frac{n}{N}}(t)Big(u_{frac{n+1}{N}}(t)-u_{frac{n-1}{N}}(t)Big), qquad nin {1, cdots, N}.
end{align}

This is equivalent to say that
begin{align}
frac{d}{dt}u_{frac{n}{N}}(t)=u_{frac{n}{N}}(t)frac{Big(u_{frac{n+1}{N}}(t)-u_{frac{n-1}{N}}(t)Big)}{-frac{2}{N}}cdotBigg(-frac{2}{N}Bigg).
end{align}

Maybe I need to scale the time variable by $N$ but the term that appears in the right hand side it looks like $ufrac{partial}{partial x}u$ after sending $Ntoinfty$.

Is my argument correct? Could someone help me to get something more rigorous?

Thank you very much.