Question

Define fn : [0,1] --> R by the

Answer 1

Answer:

The sequence of functions $\{x^{n}\}_{n\in \mathbb{N}}$ converges to the function

$f(x)=\begin{cases}0&0\leq x$.

Step-by-step explanation:

The limit $\lim_{n\to \infty }c^{n}$ exists and converges to zero whenever $\lvert c \rvert$. But, if $c=1$ the sequence $\{c^{n}\}$ is constant and all its terms are equal to $1$, then converges to $1$. Using this result, consider the sequence of functions $\{f_{n}\}$ defined on the interval $[0,1]$ by $f_{n}(x)=x^{n}$. Then, for all $0\leq x$ we have that $\lim_{n\to \infty}x^{n}=0$. Now, if $x=1$, then $\lim_{n\to \infty }x^{n}=1$. Therefore, the limit function of the sequence of functions is

$f(x)=\begin{cases}0&0\leq x$.

To show that the convergence is not uniform consider $0$. For any $n>1$ choose $x\in (0,1)$  such that $\varepsilon^{1/n}$. Then

$\varepsilon$

This implies that the convergence is not uniform.