Exercise 2.5.7

Question

Extend the result proved in Example 2.5.3 to the case $|b|<1$; that is, show $\lim \left(b^{n}\right)=0$ if and only if $-1<b<1$.

Ulisse Mini · Accepted Answer

If $|b| \ge 1$ then $\lim (b^n) 
e 0$ (diverges for $b 
e 1$).

Now for the other direction, if $|b| < 1$ we immediately get $|b^n| < 1$ thus $b^n$ is bounded.
  Since it is decreasing the monotone convergence theorem implies it converges.
  To find the limit equating terms $b^{n+1} = b^{n}$ gives $b = 0$ or $b = 1$, since $b$ is \emph{strictly} decreasing we have $b = 0$.

Exercise 2.5.7

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Exercise 2.5.7

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