Sequence without average density

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How to (if it's possible) build infinite binary sequence in that 0-elements density will not converge to any value?
For example, in $30\%_{one}\over70\%_{zero}$ random sequence (average) density (of 0-elements) converges to 70%,
or in 11010010001000010000010000001000... (I don't know the formula) density will be 100% (correct me if I'm wrong).

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$$01001100001111000000001111111100000000000000001111111111111111\dots$$