Understanding liminf of an integral function

Question

Let $$ I(x)=\int_x^\infty \frac{\theta(t)-t}{t^2}dt. $$ I know that $$ \lim_{x\to\infty} I(x)=0.$$ This is part of a proof showing that $\theta(x)\sim x$. The author concludes through some work that $$ \liminf_{x\to\infty}(I(\lambda x)-I(x))>0 $$ for some $\lambda>1$. He then claims that this contradicts the fact above. I am having trouble understanding why this is true. Here is my thought process: can I say that $$ \lim_{x\to\infty}I(x)=0 \;\;\implies\;\;\lim_{x\to\infty}(I(\lambda x)-I(x))=0\;\;\implies\;\;\liminf_{x\to\infty}(I(\lambda x)-I(x))=0$$ which gives us a contradiction, or is this not necessarily true?