Induction problem I need to prove using induction that
$$
\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3} + \cdots +\frac{1}{n(n+1)} >\frac{9n-1}{10(n+1)} \ ,\ for \ \ n \in \mathbb{N}
$$
I know how to prove it without induction, but when I try using induction I get 0>0 which is false.


I think your problem is that the RHS of your induction step should say only $$\frac{9(k+1) - 1}{10(k+2)}$$ because that is the substitution you get from the RHS of the original statement with $n=k+1$, whereas what you have is the RHS with $n=k$ and then also the next term of the series, but the series was from the LHS to begin with.
