I tried to prove this my proving the contrapositive of the statement: If $x_n$ or $y_n$ diverge, then $x_ny_n$ or $x_n/y_n$ must diverge. But, if we let $x_n = (-1)^n$ and $y_n=(-1)^{n+1}$, then we have $x_ny_n=(-1)^{2n+1}$, which converge to $-1$. And, we also have $x_n/y_n = (-1)^{-1}=-1$, a convergent sequence. Since the contrapositive is false, the original statement must be also false.
Proof verification: If $x_ny_n \rightarrow L$ and $\frac{x_n}{y_n} \rightarrow M$, then $x_n$ and $y_n$ converge.
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real-analysis
proof-verification
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2The result is indeed false. Your counterexample is correct. – 2017-01-19
1 Answers
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Yeah, it's false as you have correctly shown.
However, if we ask that the sequences be positive then the result it true.
Notice that if $a_nb_n$ and $\frac{a_n}{b_n}$ converge then the product: $a_n^2$ also converges, and in the case in which the $a_n$ are positive we deduce that $a_n$ converges. One can analogously show that $\frac{1}{b_n^2}$ converges, implying $b_n^2$ converges, and once again, if $b_n$ is positive then it must converge.