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.

Thank you for your help.