TITLE: A proof of a property of limits
QUESTION [10 upvotes]: Today during lecture my lecturer showed us this property, but provided no proof. 
If $$\lim_{n\to\infty} {d_{n+1}\over d_n} >1$$ then $$\lim_{n\to\infty}d_{n}=\infty $$
Is this property legit? (not to be disrespectful to my lecturer but he tends to make a lot of mistakes)
And if it is, what is the logic behind that property? How does it behave when the first limit tends to 1 or is less than 1?

REPLY [3 votes]: If $\lim_{n\to\infty}\frac{d_{n+1}}{d_n}>1$ then for there is a $\epsilon>0$ and $N\in\mathbb{N}$ such that $n>N$ implies $\frac{d_{n+1}}{d_n}>1+\epsilon\implies e^{d_{n+1}-d_{n}}>e^{0+\epsilon^\prime}\implies |d_{n+1}-d_n|>\epsilon^\prime $. Then 
$$
\lim_{n\to\infty}|d_{n+1}-d_n|>\epsilon^\prime
$$
But this contradicts the criterion of convergence of sequences cauchy.