Let $f_{n} \to 0$ pointwise on the interval $[-A,A]$ where $f_{n}$ are continuous and uniformly bounded. Can we show the convergence is uniform?
Thanks, any help appreciated.
Let $f_{n} \to 0$ pointwise on the interval $[-A,A]$ where $f_{n}$ are continuous and uniformly bounded. Can we show the convergence is uniform?
Thanks, any help appreciated.
No. Choose two sequences $(a_n)$ and $(b_n)$ of real numbers so that $0
Pointwise convergence does not imply uniform convergence. We have, however, Egorov's theorem, which for the given example states:
Suppose that $f_n$ are measurable and converges almost everywhere to some function $f$ on $[-A,A]$. For every $\epsilon>0$ there exists a measurable set $B_\epsilon\subset [-A,A]$ with measure (total length) $\mu(B_\epsilon) < \epsilon$, such that the convergence is uniform on $[-A,A] \setminus B_\epsilon$.