Exercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence

In this video I prove Theorem 2, which states that if the limit of the absolute values of a sequence is zero, then the sequence itself approaches zero. I solved this using both Squeeze Theorem and the precise definition of a limit. Using the squeeze theorem, I show that the minimum values that the sequence can be is the negative of the absolute values of its terms, and the maximum is the absolute values. Since both these sequences approach zero, our general sequence gets squeezed and is thus also zero.

Using the precise definition of the limit of a sequence, I show that since the limit is zero, the condition that the difference between the absolute value of the difference between any term of the sequence and the limit simplifies to just the absolute value of the terms itself, which is thus identical for both the general sequence and the absolute value of the terms. Thus, both sequences are identical and equal zero.

#math #sequences #absolutevalue #education #calculus

Timestamps:

- Exercise 2: Theorem 2: Limit of absolute value of a sequence: 0:00
- Solution 1: Squeeze Theorem: Limit is squeezed between positive and negative absolute values: 0:51
- Solution 2: Definition 2: Precise definition of a limit: 2:37
- Limit is zero so absolute value sign is identical to normal limit: 4:39
- Theorem 2 is solved: 5:40

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-bhuzhkpl
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUgfK34Voi6BvrNqt4X4BV
- Notes: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
- Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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