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Understanding the limit of a function is a fundamental concept in calculus and analysis. A function may not always give a clear answer at a particular point or may not be defined at all. However, we might be interested in the behavior of the function as we approach that point from within its domain. This is where the concept of a limit comes in.

The formal definition of a limit is tied to the idea of making the value of the function arbitrarily close to a specific value by making the input approach some point. In mathematical terms, we say that the limit of a function \( f(x) \) as \( x \) approaches \( c \) is \( L \) (denoted as \( \lim_{x\to c} f(x) = L \)) if for every positive number \( \epsilon \), however small, there exists a positive number \( \delta \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \).

This concept is crucial when analyzing the behavior of sequences of functions, especially when considering point-wise convergence, where we are concerned with the limiting behavior of a function at every single point in its domain.

A sequence of functions \( \{f_n\} \) is a list of functions \(f_1, f_2, f_3, ...\) that are usually expressed in terms of \(n\), where \(n\) is a natural number. This sequence can converge to a function \(f\) as \(n\) becomes very large. For instance, if we are given a sequence of functions \(f_n(x)\), the point at which every function in the sequence corresponds to a specific input \(x\), is of interest when we discuss convergence.

The convergence can be understood in different ways, but point-wise convergence pertains to the behavior of the sequence at each individual point. We say that a sequence of functions \( \{f_n\} \) converges point-wise to a function \(f\) on a domain \(D\) if for every point \(x\) in \(D\), the sequence \( \{f_n(x)\} \) converges to \(f(x)\). To confirm this, we check if \( \lim_{n\to \infty} f_n(x) = f(x) \) holds true for all points \(x\) in the domain of interest.

The behavior of a sequence of functions is important because it represents how a function can be approximated by a series of simpler functions, which is a common technique in mathematical analysis. However, the convergence of such sequences can be delicate and requires a careful examination of the behavior at every point, as we don't assume uniform behavior across the domain.

In the context of probability and statistics, 'convergence in probability' is a type of convergence that relates to random variables. This concept is essential when considering sequences of random variables rather than deterministic functions.

To say that a sequence of random variables \( \{X_n\} \) converges in probability towards the random variable \(X\) means that for any positive number \( \epsilon \), the probability that \(X_n\) differs from \(X\) by more than \( \epsilon \) tends to zero as \(n\) goes to infinity. Symbolically, it's written as \(P(|X_n - X| > \epsilon) \rightarrow 0\) as \(n \rightarrow \infty\).

In the case of our original exercise, we were considering a sequence of functions that were aimed to represent a random variable in the form of \(f_n(x)\). However, without the probability distribution of the underlying set \(B_{m, n}\), we cannot apply the concept of convergence in probability directly. Nevertheless, the idea is related; in deterministic functions, we look at the limit of the sequence of function values, while in stochastic processes, we consider the behavior of the probability distribution of the random variable.