Sequence of step functions satisfying $\frac{\int^1_0\phi (t)dt}{||\phi_n||_{\infty}}\rightarrow 0 \text{ as $n \rightarrow \infty$}$

Question

Is there a sequence of step functions $\phi_n :[0,1] \rightarrow \Bbb R (n \in \Bbb N)$ that satisfy $\int^1_0 \phi_n(t)dt \gt 0$ and $$\frac{\int^1_0\phi_n (t)dt}{||\phi_n||_{\infty}}\rightarrow 0 \text{ as $n \rightarrow \infty$}$$

I can't seem to conceive a function that would satisfy such conditions.

Answer 1

If $\psi_n(t)=1$ for $t \leq \frac 1 2$ and $-1$ for $t >\frac 1 2$ the $\int_0^{1} \psi_n (t)dt=0$ for all $n$ and $\|\psi_n\|_{\infty}=1$ for all $n$. Now take $\phi_n =\frac 1 n+\psi_n$. Note that $\|\phi_n\|_{\infty}\geq 1-\frac 1 n$.

Answer 2

For $n=1,2,\dots$ define $\phi_n = 1$ on $[0,1/n]$ and $0$ elsewhere. Then for each $n,$ $\phi_n$ is a step function, $\|\phi_n\|_\infty=1,$ and $\int_0^1\phi_n = 1/n.$ We're done.