Some inequalities in the paper: Acceleration of Stochastic Approximation by Averaging

Question

Suppose there exists a positive sequence $(\gamma_i)_{i\geq 1}$, which satisfies:

1. $\gamma_i \rightarrow 0$ and
2. $\frac{\gamma_i - \gamma_{i+1}}{\gamma_i} = o(\gamma_i)$,

where $o(\cdot)$ represents a sequence which converges to $0$ as $i\rightarrow \infty$. Can we prove that $$\frac{i\gamma_i}{\underset{j\leq i}{max} j\gamma_j} \geq K$$

for some absolute positive constant $K$?

Note: This question comes from Part 4 in the proof of Lemma 1 of a paper by Polyak and Juditsky in 1992. They further claimed that $m_j^i = \sum_{k=j}^i\gamma_k\geq \mu(ln(i/j))$ for $\mu$ arbitrarily large, and this claim is also not easy to understand for me.