Definition of a hard-core:
Let $h: \{0, 1\}^* \rightarrow \{0, 1\}^*$ be a poly-time-computable function s.t. $|h(x)| = |h(y)| (\forall |x| = |y|)$ and let $l(n) := |h(1^n)|$
$h$ is a hard-core of a function $f$ if $\forall$ ppt $D$, $\forall$ polynomial $p(.) \ge 0$, and all sufficiently large n's $$ |\mathbb{P}[D(f(X_n), h(X_n)) = 1] - \mathbb{P}[D(f(X_n), R_{l(n)}) = 1]| < \frac{1}{p(n)} $$ where $X_n$ and $R_{l(n)}$ are two independent random variables, the first uniformly distributed over $\{0, 1\}^n$ and the second uniformly distributed over $\{0, 1\}^{l(n)}$.
Definition of a hard-to-approximate: Let $h: \{0, 1\}^* \rightarrow \{0, 1\}^*$ and $l: \mathbb{N} \rightarrow \mathbb{N}$ satisfy $|h(x)| = l(|x|)$ for all $x \in \{0, 1\}^*$. $h$ is hard to approximate from f if for every probabilistic polynomial-time algorithm $A$, every positive polynomial $p$, and all sufficiently large $n$'s
$$ \mathbb{P}[A(f(X_n)) = h(X_n)] < 2^{-l(n)} + \frac{1}{p(n)} $$
where $X_n$ is uniformly distributed over $\{0, 1\}^n$.
It is stated that for $l(n) = O(\log n)$ and polynomial-time computable $h$, the function $h$ is a hard-core of $f$ $\iff$ $h$ is hard to approximate from $f$.
I need to prove for super-logarithmically growing $l$ one direction doesn't hold.
I suppose that if we could distinguish a function from random input, it doesn't mean we could approximate it (or if we couldn't approximate it, doesn't mean we couldn't distinguish it from random input).
But I can't come up with a proof. I'd appreciate any help, thanks.
