Q:

Continuous functions on closed intervals that aren't uniformly continuous on that interval

I am asked to give an example of sequences $(x_n)$ and $(y_n)$ in $[a,b] \subset \mathbb{R}$ such that $|x_n-y_n|<\frac{1}{n}$ but $|f(x_n)-f(y_n)|$ is bounded away from zero where $f(x)$ is continuous on $[a,b]$.
I couldn't think of an example. In fact, I wonder whether my professor made a mistake. 

A:

Suppose that $|f(x_n)-f(y_n)| \geq C>0$. Since $x_n,y_n$ are bounded sequences, they have convergent subsequences. Without loss of generality assume that $x_n \to x,y_n \to y$. The fact that $|x_n-y_n|<1/n$ implies that $x=y$. 
Now passing to the limit in  $|f(x_n)-f(y_n)| \geq C>0$ we get that $0\geq C>0$. Contradiction.

