Proof on real decimal fractions [closed]

I'm studying my first course on Real Analysis and I hoped you could help me with: Let c≥ 0 be a real number. Verify that the sequence a0, a1, a2, . . . defined by $a_n := ⌊10^nc⌋− 10⌊10^{n-1}c⌋$,a real decimal fraction. Then show that this gives rise to a bijection between $\mathbb R$ and the set of all real decimal fractions.

My textbook defines a real decimal fraction as follows: We call real decimal fraction any sequence of integers $a_0,a_1,a_2,a_3,...$, with $0 ≤ a_n ≤ 9$ for all $n ≥ 1,$ and with the property that for every $n_0 ≥ 1$ there exists an $ n ≥ n_0 $ with $ a_n \neq 9 $.