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How do I prove $$H_n − 1 \leq \ln{n} \leq H_n − \frac{1}{n}$$ where, $$H_n=\sum_{k=1}^{n}{\frac{1}{k}}$$

I was trying to do this using $f'(k + 1) ≤ f(k + 1) − f(k) ≤ f'(k)$ (proven using Lagrange's mean value theorem) and using the fact that $\sum_{k=1}^{n}{\frac{1}{k}}\leq\ln{n}$.

Scripto Meow
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1 Answers1

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Hint: $$\int_k^{k+1}\frac{1}{k+1}dx <\int_k^{k+1}\frac{1}{x}dx <\int_k^{k+1}\frac{1}{k}dx$$

NN2
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