is $f_n=x^{-n} $convergent point wise? uniformly?

Question

I believe on [1,2] $$f_n=x^{-n} $$ is point wise convergent because in n tends to infinity, there can be a function like f=0, such that $$|f_n-f|$$ tends to zero. hence this function is point wise convergent and it converges to 0; am I right? secondly, it is not uniformly convergent because when n tends to infinity , $$||f_n-f||_{\infty}$$ does not tend to zero. am I correct. I am not sure about my explanations.

Answer 1

At the point $x=1$ it actually converges to $1$. At any other point it converges to $0$. Hence the limit function is not continuous, so the convergence cannot be uniform. (uniform limit of continuous functions must be continuous)