For random variables, if two of them are non-zero correlated, are they dependent to each other? Can anyone prove this statement?
 A: Correlation and independence, in their technical senses in statistics, are conceptually different and only indirectly related.  Correlation is a combination of second central moments of a bivariate random variable while the independence concerns the joint probability distribution.  Although the relationship, as expressed in the question, is so familiar and intuitive that we take it for granted, it does merit a rigorous demonstration.
There are some subtleties.  Wikipedia gets this wrong where it states, without qualification, that independent variables are uncorrelated.  This is not always true: the implication fails when either (or both) of the variables have infinite or zero variance, for then their correlation is undefined, not zero.
I have found a thread here on CV where most of the answers make the same error.  A remarkably careful answer in a related thread uses a sophisticated characterization of independence to make and prove a correct formulation of the relationship.  Here I resort to a different approach based on conditional expectations.

Suppose $(X,Y)$ are independent. One definition is that the distribution of $X$ is the same no matter what value $Y$ might have.  This implies that the conditional expectation, if it exists, is constant, whence
$$E[X] = E[X\mid Y].$$
Using the most basic properties of conditional expectation we may deduce
$$E[XY] = E[E[XY]\mid Y] = E[E[X\mid Y]Y] = E[E[X]Y] = E[X]E[Y].$$
When the correlation $\rho(X,Y)$ exists, both variances are finite and nonzero; and all second moments of $(X,Y)$ are finite, whence $E[X]$ and $E[Y]$ exist and are finite.  A formula for the correlation is
$$\rho(X,Y) = \frac{E[XY]-E[X]E[Y]}{\sqrt{\operatorname{Var}(X)\operatorname{Var}{Y}}} = \frac{0}{\sqrt{\operatorname{Var}(X)\operatorname{Var}{Y}}} = 0.$$
This shows that

Independent variables either have no defined correlation or else their correlation is zero.

When the correlation is nonzero, this logically implies the variables are not independent, QED.
