1. ## Equicontinuity

Hi, I've been working on:
Construct a bounded sequence of continuous functions $\displaystyle f_n:[0,1] \rightarrow \mathbb{R} \\ s.t. \left| \left| f_n - f_m \right| \right| = sup \left| f_n(x) - f_m(x) \right| = 1, n \neq m, x \in [0,1]$

Can such a sequence be equicontinuous?

So far I have the example that $\displaystyle f_n(x) = sin(nx)$ but I don't know how to handle the equicontinuity. I know this particular family of functions does not have a uniformly convergent subsequence so it can't be equicontinuous, but I don't know how to rigorously prove it. Any help would be much appreciated!

2. Look at the sequence of functions $\displaystyle f_n(x) = sin(nx)$

Let $\displaystyle x_n = \frac{3\pi}{2n}$, $\displaystyle y_n = \frac{\pi}{n}$. Then $\displaystyle |x_n - y_n| = \frac{\pi}{2n}$ which converges to $\displaystyle 0$ as $\displaystyle n \rightarrow \infty$, but $\displaystyle |f_n(x_n) - f_n(y_n)| = 1, \forall n \ge 1$

Hence the sequence is not equicontinuous. Hope this helps!

3. Yes, thanks so much!