Prove that every real number \(\displaystyle{x}\in{\left[{0},\ {1}\right]}\)

Ashleigh Shaffer

Ashleigh Shaffer

Answered question

2022-03-30

Prove that every real number x[0, 1] has a "certain" binary representation

Answer & Explanation

kattylouxlvc

kattylouxlvc

Beginner2022-03-31Added 11 answers

Step 1
However, if you don't want to just copy an existing proof, you can define bi exactly like so:
Let nN be arbitrary. Now, define xn1=k=1n1bk2k
Now, define
bn={0ifxxn1<2n1ifxxn2n
and you can then prove that, by construction, xn converges to x.

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