<s>[INST] Induction proof of a known harmonic sum I want to prove that $$\frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2^n} \leq 1$$  only by induction!
I check for the first one, $\frac12 \leq 1 $ correct.
Then I assume for $n=k$ : $$\frac12 + \frac14 + \dots + \frac{1}{2^k} \leq 1$$
And Try and prove for $n=k+1$
$$\frac12 + \frac14 + \dots + \frac{1}{2^k} + \frac{1}{2^{k+1}} \leq 1$$
But I know that $$\frac12 + \frac14 + \dots + \frac{1}{2^k} \leq \frac12 + \frac14 + \dots + \frac{1}{2^k} + \frac{1}{2^{k+1}} \leq 1$$ and so:
But I am stuck, this tells me that the sum for $n=k+1$ is always $1$ , not $S \leq 1$ I am so confused, because I can't use the geometric series sum formula.. any help would be appreciated!
 [/INST] Hint : Prove the following stronger hypothesis induction :
$$\frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2^n} = 1 - \frac{1}{2^n}$$
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