Examples that the conclusion of Arzela-Ascoli does not follow when equicontinuity or uniform boundedness is dropped?

Question

I am looking for a sequence of functions that satisfies these properties:

$1$. Let $I=[0,1]$. $f_n\in C(I)$ and are continuously differentiable

$2$. $f_n(0)=0$ for every $n$

$3$. $|f_n'(x)|\le \dfrac{1}{1-x}$ for every $x\in (0,1)$

$4$. The sequence $(f_n)$ has no uniformly convergent subsequence on $I$

I would like for $f_n$ to be either not equicontinuous (when considering the sequence as a family of functions), or for them to not be uniformly bounded. I can come up with a few examples that work barring condition $(2)$, but none with all of them. Any ideas?

Answer 1

Let $f_n(x)$ be the polynomial $$f_n(x)=\sum_{k=1}^n \frac{x^k}{k}.$$

It is clear that 1) and 2) are satisfied. 3) holds because $f_n'(x)$ is a partial sum of the infinite series for $1/(1-x)$.

To prove 4), suppose for contradiction that a subsequence of $\{f_n\}$ converged uniformly to some function $g$. Then $g$ is a continuous function which vanishes on $[0,1)$, so $g(1)=0$. On the other hand, $f_n(1)\to\infty$ (because the harmonic series diverges). This contradicts subsequential uniform convergence, so property 4) holds.