Uniform convergence of sum to constant, plus $L^2$ convergence of summands to constant, implies uniform convergence of summands?

Question

If one has $f_N,g_N \geq 0$ continuous, defined over some compact sets (assumed of measure one for simplicity), such that:

1. $||f_N+g_N||_{L^1} = 1$;
2. $f_N + g_N \to C$ in $L^\infty$;
3. $f_N \to C_f$ in $L^2$;
4. $g_N \to C_g$ in $L^2$;

is it true that $f_N$ (resp $g_N$) $\to C_f$ in $L^\infty$?

Answer 1

Let $h_n(x)=nx$ for $0 \leq x \leq \frac 1 n$ and $0$ for $x >\frac 1 n$. Let $C=1,f_n=1-h_n$ and $g_n=h_n$. [Note that $0\leq h_n \leq 1$ and $\|h_n\|_{\infty}=1$ for all $n$].