<s>[INST] Proving that $\frac{1}{2} \le \frac{1}{2^n+1} + \frac{1}{2^n+2} + ... + \frac{1}{2^n + 2^n}$ I'm having trouble proving that:
$$\frac{1}{2} \le \frac{1}{2^n+1} + \frac{1}{2^n+2} + ... + \frac{1}{2^n + 2^n}$$
Edit: The next step is actually a mistake. I've put up a comment to the accepted answer explaining why it's a mistake.
I've tried reducing every single fraction on the right side to $\frac{1}{2^n + 2^n}$, but then I get result:
$$\frac{1}{2} \le \frac{1-\frac{1}{2^n}}{2}$$
which is not true.
I also tried to put this harmonic sequence in Wolfram Alpha and it tells me it equals $\ln(2)$, which is greater than $\frac{1}{2}$.
 [/INST] For any $\;1\le k\le 2^n\;$ we have that
$$\frac1{2^n+k}\ge\frac1{2^n+2^n}=\frac1{2^{n+1}}\;,\;\;\text{so:}$$
$$\underbrace{\frac1{2^n+1}+\ldots+\frac1{2^n+2^n}}_{2^n\;\text{summands}}\ge\frac{2^n}{2^{n+1}}=\frac12$$
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