Pointwise convergent but not uniform convergent function example clarification

Question

I am not completely sure why is the supremum norm of the difference of $f_n$ and $f$ is $1$? Could somebody explain what that exactly means? Is it the upper limit of their difference? On $[0,1]$, $f_n$ converges to $0$ as $n$ grows, so $0-f$ would be $-1$ and hence $|-1|$$=1$? so $f$ is $1$? I'm confused. Could you please clear it up for me?

Answer 1

$|f_n(x)-f(x)|=x^{n} <1$ if $0< x<1$ and $|f_n(x)-f(x)|=0$ if $ x=1$. Hence the supremum of $|f_n-f| $ is $1$ (because $\sup \{x^{n} :0<x<1\}=1$). For uniform convergence this supremum has to approach $0$ as $n \to \infty$.