picture.) Now you can choose $$f_n=2^{n+1}\triangle_{1/2^n,1/2^{n+1}}.$$ You get $\int f_n(x)\; \mathrm{d}x=\frac12$ and $$\int |f_n(x)-f_m(x)|\; \mathrm{d}x=\int f_n(x)\; \mathrm{d}x+\int f_m(x)\; \mathrm{d}x=1$$ for $n\ne m$. (Notice that the support of $f_n$ and the support of $f_m$ are disjoint.) share | improve this answer answered Mar 25 at 8:51 Martin Sleziak 16.8k 3 24 65