In this part you are given x and have to show the existence of the sequenced $a_n$.
So this means, for each fixed $x$ in $[0,1]$, you have to do two things:
1) show how to construct the sequence $a_n$ from the number $x$, and
2) show that the resulting sum $\sum a_n/2^n$ actually converges to $x$.
In the previous part of the question you showed such sums converge, but in this latter part of the question you are to show for each $x$ you can find an appropriate sequence which converges with its sum being that particular $x$.
EDIT: The OP Alti has asked for details on the construction of the $a_n$ from the number $x \in [0,1]$. First note that the particular number $x=1$ has the expression in which all $a_n=1$, i.e. $1=1/2+1/4+1/8+...$, so that we may assume in fact that $X \in [0,1)$, the half-open interval where $0 \le x < 1.$
To get the construction started, we use that $[0,1)=[0,1/2) \cup [1/2,1)$ where the union is disjoint. We let $a_1=0$ if $x \in [0,1/2)$ and $a_1=1$ if $x \in [1/2,1)$. Note for this "base case" of the construction that we have $a_1/2 \le x < a_1/2+1/2$, which may be restated as $x \in [a_1/2,a_1/2+1/2)$ To construct the next $a$, which is $a_2$, we use that $[a_1/2,a_1/2+1/2)=[a_1/2,a_1/2+1/4) \cup [a_1/2+1/4,a_1/2+1/2),$ the union again being disjoint. We then define $a_2=0$ if $x$ is in the first half of the above disjoint union, and $a_1=1$ if x lies in the second half of the above union.
For notation of left and right endpoints, let v_n=a_1/2+a_2/4+...+a_n/2^n$, so that $v_n$ is the $n^{th}$ partial sum of the series we are constructing. Then provided we have inductively constructed each of $a_1,a_2,...,a_n$ we have at that stage that $x \in [v_n,v_n+1/2^n).$ For constructing $a_{n+1}$ we use the disjoint union $[v_n,v_n+1/2^n)=[v_n,v_n+1/2^{n+1}) \cup [v_n+1/2^{n+1},v_n+1/2^n).$ This is another disjoint union, and we let $a_{n+1}=0$ if $x$ lies in the left half and $a_{n+1}=1$ if $x lies in the right half.
Convergence of the partial sums to x$ can be based on the nested interval theorem, or on using the partial sums of the series and also the lengths of the constructed half-open intervals in the proof.