Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$ 
Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$.

I have to prove this inequality using induction, I proved it for $n=1$ and now I have to prove it for $n+1$ assuming $n$ as hypothesis, but this seems impossible to me because the difference between the sum of $n$ and the sum of $n+1$ is a positive value. Adding a positive value on both sides of the inequality, I don't know how to prove that is always less than or equal to $\frac{3}{4}$. 

Note that you do not use induction in the above.
Another way:
Define $$S_n := \sum_{i=1}^n \frac{1}{n+i}$$
Then $$S_{n+1} = \sum_{i=1}^{n+1} \frac{1}{n+1+i} = \sum_{i=1}^{n+1} \frac{1}{n+(i+1)} = \sum_{i=2}^{n+2} \frac{1}{n+i}.$$
Hence $$S_{n+1} = S_n - \frac{1}{n+1} + \frac{1}{n+n+1} +\frac{1}{n+n+2}.$$
Now proceed...
