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I wrongly tried to prove the sequence $a_n=\frac{n}{2^n}$ is increasing. (It is obvious the sequence is decreasing but I didn't realized it at first and I was proving it is increasing by mistake!)

So I used mathematical induction to prove $a_{n}\leq a_{n+1}$:

$n=1 : a_1=\frac{1}{2} \leq a_2=\frac{1}{2}$

Assume $n=k : a_k \leq a_{k+1} $

Now we prove for $n=k+1 : a_{k+1} \leq a_{k+2}$ or $\frac{k+1}{2^{k+1}} \leq \frac{k+2}{2^{k+2}}$

We assumed $a_k \leq a_{k+1} \rightarrow \frac{k}{2^k} \leq \frac{k+1}{2^{k+1}}$

So $k (2^{k+1}) \leq 2^k (k+1)$ multiply both side by $2$ and we obtain : $k(2^{k+2}) \leq 2^{k+1} (k+1)$ then add $2^{k+2}$ to the sides: $(k+1) 2^{k+2}\leq 2^{k+1}(k+3) \leq 2^{k+1}(k+2)$ Hence: $\frac{k+1}{2^{k+1}} \leq \frac{k+2}{2^{k+2}} \rightarrow a_{k+1} \leq a_{k+2}$

So I proved the sequence $a_n$ is increasing with mathematical induction But as I said earlier it is wrong and it is decreasing actually. But why this method "mathematical induction" works perfectly well here? I am sure I did exactly the steps for that (first proved it works for $n=1$ then assumed it works for $n=k$ and conclude it works for $n=k+1$) It is really strange for me to see how mathematical induction confirm the sequence is increasing . Am I missing something?

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    $\begingroup$ $2^{k+1}(k+3)\le2^{k+1}(k+2)$? $\endgroup$ – player3236 Sep 24 '20 at 11:49
  • $\begingroup$ Ah. I messed up here! good catch $\endgroup$ – Soheil Sep 24 '20 at 11:51
  • $\begingroup$ @MathLearner $a_1= \frac{1}{2} , a_2=\frac{2}{4}$ $\endgroup$ – Soheil Sep 24 '20 at 13:38
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There is a mistake in your argument. You wrote:

$$2^{k+1}(k+3)\leq 2^{k+1}(k+2)$$

This is false. In fact $2^{k+1}(k+2)<2^{k+1}(k+3)$. So this is where your inductive argument breaks down, and doesn't prove the result.

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