Nonincreasing sequence of functions

Question

Let $f_n:[a,b]\to[0,\infty)$ be a sequence of differentiable functions such that $\lim_{n\to\infty}f_n'(t)\leq 0$. The derivative is taken with respect to $t\in[a,b]$. Then is it possible to conclude that there exists a subsequence still denoted by $f_n$ such $\{f_n\}_{n\in\mathbb{N}}$ is nonincreasing with respect to $t$.

I think it is true, but cannot see how to prove.

Can someone please help, first of all if it is realy true? If so, some argument please.

Thanks.

Answer 1

HINT: Consider the following sequence of functions on [0,1]:

$$f_n(t)=-\Big(t-\frac{1}{n}\Big)^2$$