This example illustrates the pointwise convergence of a sequence of functions $(f_n)$ defined on the interval $E = [0,1]$, with each function $f_n(x) = \frac{x}{1+nx}$. The goal is to determine the pointwise limit of $(f_n)$ as $n \to \infty$. Let’s analyze the steps:

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1. Definition of the Sequence of Functions

Each function in the sequence is defined as: $f_n(x) = \frac{x}{1+nx}, \quad \text{for } x \in [0,1].$ Here, $n \in \mathbb{N}$ (the natural numbers).

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2. Pointwise Convergence

Pointwise convergence means evaluating the limit of $f_n(x)$ for each fixed value of $x$ in the domain $E = [0,1]$.

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3. Two Cases for the Limit

To determine the pointwise limit, we calculate: $\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{x}{1+nx}.$

• Case 1: $x = 0$
  If $x = 0$, then: $f_n(0) = \frac{0}{1+n\cdot 0} = 0.$ So, the pointwise limit is: $\lim_{n \to \infty} f_n(0) = 0.$

• Case 2: $x \in (0,1]$
  If $x \in (0,1]$, the numerator $x$ is fixed and positive, while the denominator $1 + nx$ grows without bound as $n \to \infty$. Thus: $\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{x}{1+nx} = 0.$

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4. Conclusion

For all $x \in [0,1]$, the pointwise limit of $(f_n)$ is: $f(x) = \lim_{n \to \infty} f_n(x) = 0.$ This is summarized as: $f_n \xrightarrow{\text{pointwise}} f, \quad \text{where } f(x) = 0 \text{ for all } x \in [0,1].$

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Why This Happens:

• When $n$ becomes very large, the term $nx$ dominates the denominator $1+nx$, causing the fraction $\frac{x}{1+nx}$ to approach $0$.

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Let me know if you'd like further clarifications or detailed exploration!

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Related Questions:

1. What is the difference between pointwise and uniform convergence?
2. How can we generalize this result to other functions $f_n(x)$?
3. What are the implications of pointwise convergence in terms of continuity?
4. How does uniform convergence differ in behavior for this example?
5. Why does $nx$ dominate the denominator as $n \to \infty$?

Tip:

When checking for pointwise convergence, always evaluate the limit for fixed $x$ and consider any edge cases in the domain!