Let $f_n, g_n : (0, 1) → \mathbb{R}$ be the sequences of functions defined
by $f_n :=x^n$ and $g_n(x) = x^n(1-x^n)$
for $x \in (0,1)$ and $n = 1,2,…….$
Then which of the following statement(s) is(are)
true?
$(a)$ Both $(f_n)$ and $(g_n)$ converge uniformly in $(0, 1).$
$(b)$ $(f_n)$ converges uniformly in $(0, 1)$ but $(g_n)$ does not converge uniformly in $(0, 1).$
$(c)$ $(g_n)$ converges uniformly in $(0, 1)$ but $(f_n)$ does not converge uniformly in $(0, 1).$
$(d)$ Both $(f_n)$ and $(g_n)$ do not converge uniformly in $(0, 1)$
My attempt : for $f_n(x) = \begin{cases}
1 \text{ if}\ x ∈ (0,1) \\
0 \ \text{if x =0}.
\end{cases}
$
so option $1)$ will not true because $f_n(x) =x^n$ doesnot converge uniformly
option $2)$ will also not true
im doubt/confusion in option $3)$ and option $4)$
Any hints/solution will be appreciated
thanks u
Best Answer
I don't understand your attempt $f_n(x)=\begin{cases}1 &\text{ if }x\in (0,1)\\0 &\text{ if }x=0\end{cases}$. This equation is definitely wrong...
If you read carefully, then you see that you just have to decide which of $f_n$ and $g_n$ congerves uniformly.
First, you need to find the pointwise limit of $f_n$ and $g_n$. For each $x\in (0,1)$ you define $$ f(x):=\lim_{n\to\infty}f_n(x)\text{ and }g(x):=\lim_{n\to\infty}g_n(x). $$ This should be not so hard.
Next, you compute the difference of $f_n$ to $f$ and $g_n$ to $g$ with respect to the supremum norm: $$ \|f_n-f\|_\infty=\sup_{x\in(0,1)}|f_n(x)-f(x)|\text{ and }\|g_n-g\|_\infty=\sup_{x\in(0,1)}|g_n(x)-g(x)|. $$ Finally, you get $$ f_n\text{ converges uniformly to }f \Leftrightarrow \|f_n-f\|_\infty\to 0\\ \text{ and } g_n\text{ converges uniformly to }g \Leftrightarrow \|g_n-g\|_\infty\to 0. $$