Question

Let x ∈ R and ∈  > 0, and assume for simplicity that x ∈ [0, 1). Then x can be written,
x = 0.x_1x_2x_3..., i.e. x = x_1.10^-1 + x_2.10^-2 + x_3.10^-3 +.....
Define a sequence of rational numbers {q_n} (n=1, inf) by
q_n = x_1.10^-1 +......+ x_n.10^-n

We need to show that there exists N ∈ Natural numbers such that
|x - q_n|<∈ , for n>=N

Choose N ∈ Natural numbers such that 10^-N < ∈