<s>[INST] How to prove that $1-\sum_{n=1}^N \frac{2^{n-1}-1}{2^{n+1}(2^{n-1}+1)} \geq \frac{2^{N-1}+1}{2^N}$? The question is in the title. Numerical computation suggests the result is true, but I don't know how to prove it rigorously.
 [/INST] I think I got it, but my answer isn't very slick. Improvements are very welcome.
Rearranging, it suffices to show that
$$\sum_{n=1}^N \frac{2^{n-1}-1}{2^{n+1}(2^{n-1}+1)} \leq \frac{1}{2} - \frac{1}{2^N}.$$
Now, I can verify this inequality by hand for $N \in \{1,2\}$. Let $N > 2$. First observe that
$$\frac{2^{n-1}-1}{2^{n+1}(2^{n-1}+1)} < \frac{1}{2^{n+1}}.$$
In particular, for $n=1$, the LHS is $0$ while the RHS is $1/4$. Thus,
$$\sum_{n=1}^N \frac{2^{n-1}-1}{2^{n+1}(2^{n-1}+1)} + \frac{1}{4} < \sum_{n=1}^N\frac{1}{2^{n+1}} < \frac{1}{2},$$
which implies
$$\sum_{n=1}^N \frac{2^{n-1}-1}{2^{n+1}(2^{n-1}+1)} < \frac{1}{4} < \frac{1}{2} - \frac{1}{2^N}$$
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