2008-20.md

course	course_year	question_number	tags	title	year
Analysis II	IB	20	IB 2008 Analysis II	2.I.3F	2008

Explain what is meant by the statement that a sequence $\left(f_{n}\right)$ of functions defined on an interval $[a, b]$ converges uniformly to a function $f$. If $\left(f_{n}\right)$ converges uniformly to $f$, and each $f_{n}$ is continuous on $[a, b]$, prove that $f$ is continuous on $[a, b]$.

Now suppose additionally that $\left(x_{n}\right)$ is a sequence of points of $[a, b]$ converging to a limit $x$. Prove that $f_{n}\left(x_{n}\right) \rightarrow f(x)$.