Sequences of functions — a spike that destroys pointwise convergence

Let $\mathbb{N}$ denote the set of all positive integers. For $n\in\mathbb{N}$, let $f_n:[0,1]\to\mathbb{R}$ be given by $f_n(x)=n(1-nx)$ if $0\le x\le\frac1n$ and $f_n(x)=0$ if $x>\frac1n$. Which of the following statements are necessarily true?