Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 5 / b

No. Let $E=\mathbb{R}$, $x_0=0$ andThe supremum norm satisfies $\Vert f_n-f{\Vert }_{\infty }=1/n\to 0$, so there is uniform convergence to the continuous function $f$. But every $f_n$ is discontinuous at zero: $f_n(0)=1/n$ whereas $f_n(x)=0$ for every $x\ne 0$. No tail consists of functions continuous at $x_0$.