0
$\begingroup$

proof

The above link is the proof of $\lim \inf x_n + \lim \inf y_n \le \lim \inf (x_n+y_n)$ when $x_n$ and $y_n$ are bounded.

Could you tell me why $XY_n \subset X_n+Y_n$??

Also, my textbook gives me a hint that 'Find a subsequence $\{x_{n_i}+y_{n_i}\}$ that converges. Then find a subsequence $\{x_{n_{m_i}}\}$ of $\{x_{n_i}\}$ that converges. Then apply what you know about limits.'. I really have no idea how to use this hint. Please tell me if you know how to use this hint.

Thank you in advance.

$\endgroup$
  • $\begingroup$ Are you using Basic Analysis by Jeril Lebl? $\endgroup$ – Atif Farooq Nov 3 '18 at 18:02
2
$\begingroup$

For the first question: For $k\geq n$, $x_{k}+y_{k}\in X_{n}+Y_{n}$ because each $x_{k}\in X_{n}=\{x_{l}: l\geq n\}$ and $y_{k}\in Y_{n}=\{y_{l}: l\geq n\}$. So we conclude that $XY_{n}\subseteq X_{n}+Y_{n}$.

For the second question: Let $\liminf(x_{n}+y_{n})=\lim_{k}(x_{n_{k}}+y_{n_{k}})$. Since $(x_{n_{k}})$ is bounded, then there is a further subsequence $(x_{n_{k_{i}}})$ of $(x_{n_{k}})$ such that $\lim_{i}x_{n_{k_{i}}}$ exists. Since $\liminf x_{n}$ is the smallest limit point of its convergent subsequences, so $\liminf x_{n}\leq\lim_{i}x_{n_{k_{i}}}$. Now the corresponding subsequence $(y_{n_{k_{i}}})$ of $(y_{n_{k}})$ need no converge, but there is a further subsequence $(y_{n_{k_{i_{l}}}})$ of $(y_{n_{k_{i}}})$ such that $\lim_{l}y_{n_{k_{i_{l}}}}$ exists. Once again we have $\liminf y_{n}\leq\lim_{l}y_{n_{k_{i_{l}}}}$. The corresponding subsequence $(x_{n_{k_{i_{l}}}}+y_{n_{k_{i_{l}}}})$ of $(x_{n_{k}}+y_{n_{k}})$ is convergent and $\lim_{l}(x_{n_{k_{i_{l}}}}+y_{n_{k_{i_{l}}}})=\lim_{k}(x_{n_{k}}+y_{n_{k}})$. But $\lim_{l}(x_{n_{k_{i_{l}}}}+y_{n_{k_{i_{l}}}})=\lim_{l}x_{n_{k_{i_{l}}}}+\lim_{l}y_{n_{k_{i_{l}}}}=\lim_{i}x_{n_{k_{i}}}+\lim_{l}y_{n_{k_{i_{l}}}}$, the result follows.

$\endgroup$
  • $\begingroup$ For the first question, why $XY_n = X_n+Y_n$does not hold?? $\endgroup$ – Sihyun Kim Mar 25 '18 at 8:47
  • $\begingroup$ $X_{n}+Y_{n}$ may contain an element of the form $x_{n}+y_{n+1}$ which is not contained in $XY_{n}$. $\endgroup$ – user284331 Mar 25 '18 at 8:51
  • $\begingroup$ Could you tell me why $y_{n_{k_i}}$ need no converge?? $\endgroup$ – Sihyun Kim Mar 25 '18 at 8:59
  • $\begingroup$ And if we follow the last line of the proof, isn't it concluded as $\lim \inf x_n + \lim \inf y_n = \lim \inf (x_n+y_n)$? $\endgroup$ – Sihyun Kim Mar 25 '18 at 9:02
  • $\begingroup$ No, because we have $\lim_{l}...\geq\liminf y_{n}$ for example. $\endgroup$ – user284331 Mar 25 '18 at 16:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.