I have found the following very useful theorem and I would appreciate some help comprehending it fully.

Theorem  Let $\{X_n \} $ be a sequence of random variables bounded in probability and let $ \{Y_n \} $ be a sequence of RVs which converge to $0$ in probability. Then

$$X_n Y_n \rightarrow 0 \quad \text{in probability} $$ 

And its proof:

For a given $\epsilon$ since $X_n$ is bounded in probability we can choose $N$ and a constant $B$ such that 

$$n \geq N \Rightarrow P \left[ |X_n| \leq B \right] \geq 1-\epsilon$$

Then

$$ \overline{\lim_{n \to \infty}}P \left[|X_n Y_n| \geq \epsilon \right] \leq \overline{\lim_{n \to \infty}} P \left[ |X_n Y_n| \geq \epsilon, |X_n| \leq B \right]+ \overline{\lim_{n \to \infty}}P \left[|X_n Y_n| \geq \epsilon, |X_n| &gt; B \right] \leq \overline{\lim_{n \to \infty}} P\left[|Y_n| \geq \epsilon/B \right] +\epsilon=\epsilon  $$

If someone could help me understand how the last inequality is derived, given the information we have, I would be grateful. I understand the second to last inequality comes from the subadditivity property of limitsuperior so I merely need to comprehend the last one. Thank you in advance.

EDIT: I think it is because $$P(A) \times P(B|A) \leq P(A) $$ and the complement of the event that is implied by the bound in probability.


Question: What lightweight telephoto lens are available for Nikon bodies?
Answer: Really you seem to have done your research already. There are a few more similar lenses, from Sigma and Tamron but they will not save you much weight and certainly not improve on image quality and AF-speed. As you noticed, lightweight and quality telephoto do not mix.

What is left is for you to consider the total weight of your system instead. Maybe you can replace more than one lens with a longer zoom ratio one. Quality is not top-notch but still can be good enough given the constraints. Something like the Nikkor AF-S 28-300mm F/3.5-5.6 ED VR maybe.