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JH
Numerade Educator

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Problem 87 Medium Difficulty

Prove Theorem 6.
[Hint: Use either Definition 2 or the Squeeze Theorem. ]

Answer

If $\lim _{n \rightarrow \infty}\left|a_{n}\right|=0,$ then $\lim _{n \rightarrow \infty} a_{n}=0$

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Video Transcript

the're, um six states, the phone. If the limit of the absolute value and zero, then we must have that the limited and is also zero. Now let's go ahead and provide a proof of this fact. So note that the absolute value of Anne is bigger than an and Anne is also bigger than the negative of the absolute value This is for all in. So now let's take a limit as n goes to infinity on all three sides. So on the left hand side, the outermost side over here we can write This is negative, Lim absolute value a n so negative zero which is just zero from the given information. We know that this one also goes to zero. So by the squeeze their own the limit of this term in the middle has to be equal to the common value here. And since the lower and upper bounds or both zero, we conclude that the limited and is also zero, and that completes the proof