Consider two real-valued sequences $\left \{ x_{n} \right \}$ and $\left \{ y_{n} \right \}$ satisfying the condition $x^{3}_{n} - y^{3}_{n} \rightarrow 0$ as $n \rightarrow \infty $. Then, as $n \rightarrow \infty $,
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$x_{n} – y_{n} \rightarrow 0$ always
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$x_{n} – y_{n} \rightarrow 0$ only if $\left \{ x_{n} \right \}$ converges
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$x_{n} – y_{n} \rightarrow 0$ only if $\left \{ |x_{n}| - |y_{n}| \right \}$ converges
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$x_{n} – y_{n} \rightarrow 0$ only if $\left \{ |x^{2}_{n} +x_{n} y_{n} + y^{2}_{n}| \right \}$ converges
