Can Positive Values of Observations imply Population Positive Mean? Suppose that we have N-observations for a random variable $V$ from a population:
$$\{v_1,\;\ldots,\;v_N\}$$
, and all of the observations is positive: $v_i>0\quad \forall i=1,\;\ldots,\;N$.
Here, assume that the observations are "randomly sampled" (i.i.d.)
I think, from the positive values of observations, it is reasonable to guess that the population mean $\mathbb{E}[V]$ is also positive because the observations may represent the population distribution due to the random sampling assumption.
But, I am also skeptical about whether a statement for data can imply a statement for a population.
So, is it possible to derive $\mathbb{E}[V]>0$ from $v_i>0\quad \forall i=1,\;\ldots,N$?
In other words, can we mathematically prove the following implication?
$$v_i>0\quad \forall i=1,\;\ldots,N \quad \Longrightarrow \quad \mathbb{E}[V]>0$$
(Note that I am not asking an inference question. I want to know whether we can prove that under the random sampling assumption.)

PROVE? NO
set.seed(21)
N <- 3 # Sample size
x <- rnorm(N, -0.1, 1)

I get three positive numbers, resulting in a positive sample mean, yet the population mean is $-0.1$.
EDIT
Being more formal with mathematics, we could calculate the distribution of the first order statistic (smallest value) of a random sample drawn from this $N(-0.1, 1)$, and we could do that for any sample size. Doing so will reveal that this gets less probable as the sample size increases, but its probability never drops to zero for any finite sample size.
(You might find the order statistic calculation to be more pleasant by using something like $U(-0.2, 0.1)$.)
