Exercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence
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0.000 HBDExercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence
by Math Easy Solutions on Math Easy Solutions View my bio on Blurt.media: https://blurt.media/c/mes [](https://blurt.media/w/ha56vYc8GGs9sHhs7JduyS) 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 . ------------------------------------------------------ Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps *** [**Check out today's video..!**](https://blurt.media/w/ha56vYc8GGs9sHhs7JduyS)