Is this logically valid? $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$
and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$
assuming that $n \in \mathbb {N}$.

I think the hidden agenda here is to prove the statement by induction.  If this is true, the inductive hypothesis given in the OP is incorrect, and instead should be 
$$1+\frac{1}{2}+\cdots+\frac{1}{n-1}>\ln \color{red}{(n-1)}$$
To build a proper induction, you begin with the above, add $\frac{1}{n}$ to both sides, then try to prove that $$\ln (n-1) + \frac{1}{n}\ge\ln (n)$$
