Find a sequence of step functions $ψ_n : [0, 1] → R$ $(n ∈ N)$ that satisfy $\int_0^1 ψ_n(t) \, dt >0$ and
$$\int_0^1 ψ_n (t)\,dt/\|ψ_n\|_∞ → 0 \text{ as } n→∞$$
I'm not really sure how to start here. From logic, these criteria imply that the area under the curve of the step functions need to be positive but can't infer much other than that. I appreciate it's a simple question but any help is welcome!
Does this $\psi_{n}(x)=\chi_{[0,1/n^{2}]}(x)$ do the job?