can someone explain this part "|x - z| + |z - y|" is used in this proof?
pic related the middle of the inequality
really i only just don't understand this exact part
did you try drawing the thing? goddamit
P is the set of points that are at a distance less than epsilon from x.
so z in P means |z-x| < epsilon
same for Q and y. z in Q means |z-y| < epsilon
the way i am understanding it now is that: if c is in U intersection V then by the definitions in my book i get |a - c| < epsilon where epsilon > 0 so |a - c| = 0 by another definition in my book so a = c.
this means that a belongs to the neighborhood of a
then . . . c belongs to the neighborhood of b also
what does the contradiction mean exactly