Do you know BERNOULLI's INEQUALITY? .
That gives one ‘side’ of the inequality.
Using Napper inequality we can get .
Using the exponential we get
What a slick way to get the lower bound. I normally induct using the binomial expansion or the identity to derive in order to show the sequence is monotonic increasing. Then it follows that since it is the first term. Let's see if I can use your method.
Let denote the sequence and let . Assume is such that . Then contradicts that is a lower bound.
Next assume There is some in the sequence such that otherwise is the greatest lower bound. But no such exists since it was proved that By Trichotomy,