For any real x, and real e>0, define U = (x-e,x+e).
Let y in U. Will prove that U is a neighborhood of y.
Given y, let rho = min( |y-(x-e)|, |y-(x+e)| ). Note that rho > 0, because... . Let V = (y-rho, y+rho). Clearly V is an open neighborhood y.
Claim V is a subset of U:
Let z in V. Then.... and so z>x-e. Also, from ... get that z<x+e. Therefore z in U, proving that V is a subset of U.
Thus have shown that any element y of U is contained in an open neighborhood V that is entirely contained in U, and hence that U is neighborhood of each of its elements.