# Proving if \$a_{k}ge a_{k-1}+1\$ then \$1+frac{1}{a_{0}}(1+frac{1}{a_{1}-a_{0}})…(1+frac{1}{a_{n}-a_{0}})le prod_{k=0}^{n}(1+frac{1}{a_{k}})\$

I’ve worked on this problem with the sequence $$a_{k}$$ being the natural numbers, that is $$a_{k}=a_{k-1}+1$$ and $$a_{0}=1$$. Over the naturals, $$prod_{k=0}^{n}(1+frac{1}{a_{k}})$$ can be proven to be $$n+1$$.

I’ve been able to recognize the obvious pattern in $$prod_{k=2}^{n}(1+frac{1}{k-1})$$ to be equal to $$n$$, but I do not know how to prove this rigorously.

That said, over the naturals, the inequality reduces to $$n+1 le n+1$$. That got me thinking if the sequence in question could be expanded with the condition $$a_{k}ge a_{k-1}+1$$ and specifically if this would not lead to the situation corresponding with RHS being greater than LHS.