It is known that the Nagumo equation

$\partial u/\partial t+D{\partial}^{2}u/\partial {x}^{2}+f\left(u\right)=0$ has a so-called travelling wave front. This means that there exists a function

$U$ such that

$U(-\infty )=0$,

$U\left(\infty \right)=1$ and that

$u(x,t)=U(x/\sqrt{D}+ct)$,

$c>0$ is a solution. In this paper so-called “discrete Nagumo equation” is considered. In fact it is an infinite system of ODE’s of the form

${\dot{u}}_{n}=d({u}_{n-1}-2{u}_{n}+{u}_{n+1})+f\left({u}_{n}\right)$,

$n\in \mathbb{Z}$,

$d>0$. The author proves (in a rigorous way) that under certain conditions on

$f$ a similar travelling wave front exists also for the discrete case. Namely it is proved that there exists a function

$U$, satisfying the conditions

$U(-\infty )=0$,

$U\left(\infty \right)=1$,

$U\left(x\right)>0,$ $\forall x\in \mathbb{R}$, and such that

${u}_{n}\left(t\right)=U(n+ct)$,

$c>0$, is a solution of the discrete Nagumo equation, provided that

$d$ is large enough. It is to be stressed that the proof given here has a clear approximational aspect.