Are two limits of probabilities of "random variable is equal to zero" and "random variable bigger than any eps>0" are complementations of each other?

Question

I'm studying some theorems in theory of probability and in some of them is used such an expression.

$$Let\space\{\xi_n, n \in \mathbb{N}\}$$ is a sequence of non-negative random variables, than if: $$\forall \epsilon > 0\lim_{n \to \infty} P(\xi_n >\epsilon) = 0 $$ than something...

Is it correct to replace this expression with this: $$Let\space\{\xi_n, n \in \mathbb{N}\}$$ is a sequence of non-negative random variables, than if: $$\lim_{n \to \infty} P(\xi_n = 0) =1 $$ than something...

My thoughts: It is correct because probability of being equal to zero and probability of being greater than any positive eps > 0 intuitively are complementations of each other. But intuition is a bad science.

Answer 1

it's trivially false; consider e.g. $\xi_n=1/n$ (degenerate random variables).