# Uniform convergence - Is this statement true? Why?

Let $M$ = subset of $\mathbb{R}$. Let sequence of functions $f_n(x)$ converges uniformly to $f(x)$ defined on $M$. Now we choose nonempty $N$ = subset of $M$. Then $f_n(x)$ also converges uniformly to $f(x)$ on $N$.

My solution:

Uniform convergence definition $(\forall \varepsilon >0)(\exists a_0\in A):a\in A\wedge a\ge a_0\wedge x\in M\Rightarrow |f_a(x)-f(x)|<\varepsilon$

We have $N$ = subset of $M$ then $x$ above is $x\in N$, so it holds: $(\forall \varepsilon >0)(\exists a_0\in A):a\in A\wedge a\ge a_0\wedge x\in N\Rightarrow |f_a(x)-f(x)|<\varepsilon$

• What have you tried? What is the definition of uniform convergence on the sets $M$ and $N$, and how are they related? – Jakob Oesinghaus Sep 30 '13 at 7:22
• I have just added my solution – Anakin Sep 30 '13 at 7:40
• @Anakin Why, looks fine... – W_D Sep 30 '13 at 7:43

The set of all continuous functions on $M$ comes with a natural norm—a way of measuring "distance from the zero function." The norm $\|f\|_M$ of a continuous function $f$ on $M$ is defined as the supremum of $|f|$ over $M$: the smallest number greater than or equal to $|f(x)|$ for all $x \in M$ (which may be "$\infty$").

Given two continuous functions $f$ and $g$, you can think of $\|f - g\|_M$ as measuring a "distance between $f$ and $g$."

A sequence $f_n$ of continuous functions on $M$ converges uniformly to $f$ if and only if the sequence $f_n$ converges to $f$ according to the $\|\cdot\|_M$ distance. In other words, as $n$ increases, the distance $\|f_n - f\|$ eventurally drops below every positive number and stays there. (Think about why this is equivalent to the definition of uniform convergence that you already know.)

Now, define $\|f\|_N$ as the supremum of $|f|$ over $N$. Observe that $\|f\|_N \le \|f\|_M$ for all continuous functions $f$ on $M$. It follows pretty easily that if the sequence $f_n$ converges to $f$ according to the $\|\cdot\|_M$ distace, it also converges to $f$ according to the $\|\cdot\|_N$ distance.

I find this phrasing useful because it lets you break the definition of uniform convergence into two conceptually meaningful pieces: the definition of the $\|\cdot\|_M$ distance and the definition of convergence according to a distance function.

For more details, see this recent answer of mine, which just happens to be on the same topic.