I'm trying to show that $\sum_{i=1}^n ln(1+\frac{1}{i}) \leq \sum_{i=1}^{n}\frac{1}{i}$ for all $n\in\mathbb{N}$. My first instinct is to go the route of saying that it is sufficient to show that $\ln(1+\frac{1}{i}) \leq \frac{1}{i}$ for all $i\in\mathbb{N}$. So first off, is this valid (as an axiom, or the result of a basic enough proof that I don't need to prove this statement)?
Assuming that that's OK, my next challenge is to prove the latter statement. It's obvious on its face, but that's not rigorous. My instinct then is to use an initial value + derivative:
- show that the left side is less than the right side for $i=1$
- show that the derivative of the left side is less than the derivative of the right side
- therefore the left side must always be less than the right side.
I'm not sure if I can use this though for two reasons - firstly, because I'm not sure what this theorem is called (the initial value + derivative theorem?) and I'm not sure it's basic enough to use without a proof. And secondly, I'm not sure you can use calculus, which is continuous, to prove properties of discrete numbers. So, maybe I'm off here. Am I alright on this track, or if not, does anyone see a better way to go about proving this?