Barbalat's Lemma Proof Typo and Clarification

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I am trying to study the proof of Barbalat's Lemma by Hao Liu as shown in this link

However, I realized that there is a typo in there:

$$\lim_{t\rightarrow\infty}\left|f\left(t_n + \delta \right)- f\left(t_n\right)\right| = \lim_{t\rightarrow\infty}|f\left(t_n + \delta \right)- \lim_{t\rightarrow\infty}\left|f\left(t_n\right)\right|$$

Shouldn't it be:

$$\lim_{t\rightarrow\infty}\left|f\left(t_n + \delta \right)- f\left(t_n\right)\right| = \lim_{t\rightarrow\infty}\left|f\left(t_n + \delta \right)\right|- \lim_{t\rightarrow\infty}\left|f\left(t_n\right)\right|$$

Also, why is the second equation valid? Can one give me a proof of that?

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There are three typos in the line:

  • The limit should be as $n \to \infty$, not as $t\to\infty$.
  • There is not a missing bar - there is an extra bar.
  • The next step should be $|\alpha - \alpha|$, not $|\alpha| - |\alpha|$.

The line should be $$\begin{align}\lim_{n\to\infty}\left|f(t_n + \delta)- f(t_n)\right| &= \left|\lim_{n\to\infty}\big(f(t_n + \delta)- f(t_n)\big)\right| \\ &= \left|\lim_{n\to\infty} f(t_n + \delta)- \lim_{n\to\infty}f(t_n)\right|\\ &=|\alpha - \alpha| \\ &= 0\end{align}$$

The interchange of $|\phantom{x}|$ and limit is justified because the absolute value is continuous.