Example that the Lebesgue integral of a function can take any value between $[0,c]$ Latest homework question in measure and integration theory course. Suppose that $f_n : \mathbb{R} \to \mathbb{R}$ are integrable, $f_n ≥0$ and $f_n(x) \to f(x)$ for every $x $. Suppose that
$$\int f_n\to c≥0 .$$ Show that $\int f$ exists
and  $0≤ \int f ≤ c$. Show by examples that every value in $[0 ，c]$ is possible.
I can show $\int f$ exist by showing $\int|f|$ being finite and use Fatou's lemma to show $\int f ≤ \liminf \int f_n = c$. 
But I can't do the part that $\int f ≥ 0 $. I am also confused since $\int f =\int\lim f_n $， how could I find an example that the Lebesgue integral is not fixed.

By pointwise comparison you have that $0 \leq \liminf f_n(x) = \lim f_n(x) = f(x)$ so that $\int f \geq 0$. 
To show that every value in $[0,c]$ is possible, we need to construct a different $f$ (and hence a different $(f_n)$) for each $\lambda\in[0,c]$. We do so as follows: 
Let 
$$ g_n(x) = \begin{cases} n & x\in (0,1/n] \\ 0 &\text{otherwise} \end{cases} $$
and let
$$ h(x) = \begin{cases} 1 & x\in [-1,0) \\ 0 &\text{otherwise}\end{cases} $$
Observe that $\int g_n \mathrm{d}x = 1 = \int h\mathrm{d}x$. 
Let $$f_{n,\lambda} = (c-\lambda) g_n(x) + \lambda h(x)$$
clearly 
$$ \int f_{n,\lambda} = c $$
so 
$$ \liminf_{n\to\infty} f_{n,\lambda} = c$$
But pointwise you have that
$$ f_\lambda(x) := \lim_{n\to\infty} f_{n,\lambda}(x) = \lambda h(x) $$
using that the pointwise limit
$$ \lim g_n(x) = 0 $$
for any $x$. This implies that
$$ \int f_\lambda(x) \mathrm{d}x = \lambda \in [0,c] $$
as desired. 
