Prove if ${x_n}$ and ${y_n}$ are Cauchy and $x_n + y_n > 0$, for all natural n, then $\frac{1}{x_n + y_n}$ cannot converge to zero.

Attempt: Suppose $x_n → a$ and $y_n → b$. Then $x_n + y_n → a + b$. Then since the sequences are Cauchy, they converge. Then $x_n$ and $y_n$ are bounded. Thus, there is a positive M real number such that $|x_n + y_n| < M$ . Then, $|1/(x_n + y_n)| < 1/M$. I am assuming it cannot converge to zero since zero is not defined in the denominator.

Can someone please help me finish? Thank you very much.


You are basically done. Since $(x_n)$ and $(y_n)$ are Cauchy, they are bounded. In particular $(x_n + y_n)$ is bounded, so there is some constant $M$ such that $|x_n + y_n| < M$ for all $n$. Then (and here is a little mistake in your attempt), we have $\left|\frac1{x_n + y_n}\right| > \frac1M$ for all $n$.

Now if $\frac1{x_n + y_n}$ did converge to $0$, then we could make its absolute value as small as we want by choosing a very large $n$. However, the above relation shows that this absolute value is always bigger than $\frac1M$.


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