I am using Spivak calculus. Now Iam in the chapter limits. In pages 97-98, he has given the example of Thomaes function. What he intends to do is prove that the limit exists. He goes on to define the thomae's function as f(x)=1/q, if x is rational in interval 0<x<1 here x is of the form p/q where p and q are integers and q>0 f(x)=0, if x is irrational He proves that the limit of f(x) as x tends to a(it is between 0 and 1) is 0 a can be either rational or irrational He starts the arguement like this. He assumes a natural number n so large such that (1/n)<=€, ie..(1/n) is lesser than or equal to epsilon. Since, L =0 ( The assertion he gives on value of this limit), he writes that |f(x)-L| or |f(x)-0|<€, He says that the numbers x which dont satisfy this inequality are... 1/2, 2/3 1/4, 3/4, 1/5,.. and he also includes 1/n....(n-1)/n.. My doubt is how? Why dont these numbers satisy/ why is it false? For example, consider x=1/n so, f(x)=1/n |1/n|<€. Is it not true? Isn't the inequality true? Wont it lead to contradiction? Where have I gone wrong?