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I'm doing Problem II.5.2 in textbook Analysis I by Amann/Escher.

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Could you please verify whether my attempt on $\underline{\lim} (x_n y_n) \le x_* y^*$ is fine or contains logical gaps/errors? Any suggestion is greatly appreciated!


My attempt:

We have $$\begin{aligned} \underline{\lim} (x_n y_n) &= \lim_{n \to \infty} \left(\inf _{k \geq n} (x_k y_k) \right) &&\le \lim_{n \to \infty} \left (\inf _{k \geq n} \left (x_k \sup _{k \geq n} y_k \right)\right) \\ &= \lim_{n \to \infty} \left ( \sup _{k \geq n} y_k \left ( \inf _{k \geq n} x_k \right)\right) &&= \lim_{n \to \infty} \left ( \sup _{k \geq n} y_k \right) \lim_{n \to \infty} \left ( \inf _{k \geq n} x_k \right) \\ &= \overline{\lim} (y_n) \underline{\lim} (x_n) &&= y^* x_* \\ &= x_* y^*\end{aligned}$$ This completes the proof.

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It looks fine to me, the only thing I would suggest is to use different notation after your inequality: $$\lim_{n\rightarrow\infty}\left(\inf_{k\geq n}\left(x_{k}\sup_{i\geq n}y_{i}\right)\right)$$ This to avoid confusion concerning what you are taking an infimum of.

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