Proof Verification: Convergence

Question

Prove that $b^n \to 0$ as $n \to \infty$ for $ 0<b<1$

Proof by the ratio test:

$b^{n}/b^{n-1}$ = $b$ $ \to b<1$

$ \implies b^n \to 0$ as $ n \to \infty$

Is this proof correct? Can anyone please verify it?

Answer 1

Your proof is correct with slight modification.

You meant $$b^{n+1}/b^{n}=b\to b$$

$$b<1\implies $$

$$ lim _{n\to \infty} b_n =0$$

Which is correct by the ratio test for positive sequences.

Answer 2

Your proof isn't correct, because besides the monotonicity of your sequence you should also prove its boundness. As an another way, you can use the fact $b^n = e^{n \ln b}$ and $\ln b < 0$ hence $e^{n \ln b} \to 0$

Answer 3

$$0 <b <1 \implies \frac {1}{b}>1$$

put $$\frac {1}{b}=1+\epsilon . $$

Using the fact that $$(1+\epsilon)^n\ge 1+n\epsilon $$ which we prove by induction, we get $$\lim_{n\to+\infty}\frac {1}{b^n}=+\infty $$ and $$\lim_{n\to+\infty}b^n=\frac {1}{+\infty}=0.$$