A proof of a property of limits

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?

Assume that this is a positive sequence. (You might have $\lim_{n\to \infty} d_n = -\infty$). There is a $M$ and $\delta > 0$ such that for $n\geq M$ $$ d_{n+1}/d_n > 1 + \delta = a > 1. $$ That is: $$ d_{n+1} > ad_n. $$ So for $n> M$: $$d_n > ad_{n-1} > a^2d_{n-2}... > a^{n-M}d_M.$$ Now let $n\to \infty$

Since we have $\lim_{n \to \infty } d_{n+1}/d_n > 1$,

Let us have $ \delta = \min \{ d_{n+1} - d_n: n\ge N \text{ for some N }\in \mathbb N\}$, then we have $ \lim_{n \to \infty} d_n > \lim_{n \to \infty} d_N + n\delta$ which diverges to $\infty $.