Non puntual sequences are not an equicontinous family

Question

Given a sequence $u_{n}$ of (non constant) functions in $\mathcal{C}(\overline{B},\mathbb{R})$ with $\overline{B}$ the closed unit ball in $\mathbb{R}^{d}$. Then, is it true that if $u_{n}$ does not converge point-wise then it can't be equicontinous?

Answer 1

Equicontinuity and boundedness imply a (uniformly) convergent subsequence. But equicontinuity does not imply pointwise convergence: If $a_n$ is a non-convergent sequence of reals, then the functions $f_n(x)=x+a_n$ are equicontinuous but do not converge pointwise.