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You can show it's cauchy by noting

|x_n - x_m| =< |x_n - x_(n-1)| + ... + |x_(m+1)-x_m|

< 1/2^(n-1) + ... + 1/2^m

< 1/2^(m-1) -> 0 as m,n -> ∞

|x_n - x_m| =< |x_n - x_(n-1)| + ... + |x_(m+1)-x_m|

< 1/2^(n-1) + ... + 1/2^m

< 1/2^(m-1) -> 0 as m,n -> ∞

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