Prove that if $\lim_{n\to\infty} \frac{x_n}{y_n} = 0$, then $\lim_{n\to\infty} x_n \div \frac{x_n + y_n}{2} = 0$ Let $\{x_n\}$ and $\{y_n\}$ be positive sequences. Assume $\lim_{n\to\infty} \frac{x_n}{y_n} = 0$. I have to prove if the claim $\lim_{n\to\infty} x_n \div \frac{x_n + y_n}{2} = 0$ is always true, always false, or sometimes true and sometimes false depending on the sequences.
My attempt:
\begin{align*}
x_n + y_n &\geq y_n \\
0 < \frac{1}{x_n + y_n} &\leq \frac{1}{y_n} \\
0 < \frac{x_n}{x_n + y_n} &\leq \frac{x_n}{y_n}
\end{align*}
Then by Squeeze Theorem, $\lim_{n\to\infty} \frac{x_n}{x_n + y_n} = 0 \implies \lim_{n\to\infty} \frac{2x_n}{x_n + y_n} = 0$ which is what we wanted. Could you check if my proof makes sense. Is there any cases of sequences of x_n and y_n such that this statement is not true?

$$\frac{x_n}{x_n+y_n\over 2}=2\frac{x_n \over y_n}{{x_n\over y_n}+1}$$
Therefore $$\lim_{n\to \infty}\frac{x_n}{x_n+y_n\over 2}=2 \frac{0}{{0}+{1}}=0 $$
