bounded convergence theorem is not holding

Question

Consider the example $$f_n(x)=\cases{ 0 & if $| x| \ge {1\over n}$\cr n\Bigl(1+nx\Bigr) & if ${-1\over n}\leq x \leq 0 $\cr n\Bigl(1-nx\Bigr) & if $ 0 \le x \le \frac{1}{n}$}$$ Each $f_n$ is piecewise linear and continuous on the interval $[0,1]$, and $f_n(x)\to 1$ as $n\to\infty$ for each $x\in(0,1]$. However, $\int_0^1 f_n(x){\rm d}x=1$ for each $n$, so the conclusion of the Bounded Convergence Theorem does not hold here. So problem should be this sequence of functions is not uniformly bounded?