91Ó°ÊÓ

Pointwise convergence is a fundamental concept within the field of real analysis, specifically when examining the behavior of sequences of functions. To put it simply, a sequence of functions \( \{f_n(x)\} \) is said to exhibit pointwise convergence to a function \( f(x) \) on a set \( I \) if, for each individual point \( x \) within that set, the values of \( f_n(x) \) get arbitrarily close to \( f(x) \) as \( n \), the index of the sequence, grows without bound.

Imagine standing at each point \( x \) on the interval \( I \) and watching the values of the sequence \( f_n(x) \) as \( n \) increases. If these values settle down to the value of \( f(x) \) at that specific point, and stay close to it as \( n \) continues to increase, then we have pointwise convergence at \( x \) on \( I \). Using the principle of pointwise convergence, the step by step solution shows that for the given sequence, regardless of which \( x \) you're standing at within the interval \( [0, 1] \), the sequence \( f_n(x) = nxe^{-nx^2} \) approaches the function \( f(x) = 0 \) as \( n \) becomes larger and larger. This is confirmed by evaluating the limit of \( f_n(x) \) as \( n \rightarrow \) infinity, which indeed equates to zero for all points on the interval.

A sequence of functions can be best understood as a list of functions ordered by an index, commonly denoted by \( n \) that typically runs through the set of natural numbers. These functions are often related in some manner, creating a progression as \( n \) increases. In our specific exercise, we have a sequence defined by \( f_n: x \rightarrow n x e^{-n x^{2}} \) for each natural number \( n \).

When examining sequences of functions, we look at their behavior as \( n \) becomes large; what they approach (\