Let $\{f_k\}_{k \in \mathbb{N}}$ be a family of real positive measurable function on $D \subseteq \mathbb{R}$ such that $0 \leq f_0 \leq f_1 \leq \ldots $. If $\lim f_k = f$ on $D$ then $$\lim \int_D f_k(t)dt = \int_D \lim f_k(t) dt = \int_D f(t) dt.$$
Is this proof is right? And is there another way for the solution? Thank you!
PROOF. Let $\{g_k\}_{k \in \mathbb{N}}$ be a sequence of non-negative measurable functions on $D$. If $\lim_{k \to +\infty} g_k = g$ then we have $$\int_D g = \int_D \lim g_k = \int_D \liminf_{k\to +\infty}( g_k )\leq \liminf_{k \to + \infty} \left(\int_D g_k \right) = \lim_{k \to + \infty} \int_D g_k.$$ And then $$\int_D g \leq \lim_{k \to + \infty} \int_D g_k.$$ Note that $0 \leq f_0 \leq f_1 \leq \ldots \leq f$ on $D$. Then
$\bullet$ Let us consider the family $\{f - f_k\}_{k \in \mathbb{N}}$. We have $\lim_{k \to +\infty } (f - f_k ) = 0$ and $f - f_k$ is a non-negative function for any $k \in \mathbb{N}$. Then $$ 0 \leq \lim_{k \to + \infty} \int_D (f - f_k)$$ or $$ \lim_{k \to + \infty} \int_D f_k \leq \int_D f.$$
$\bullet$ Let us consider the family $\{f_k\}_{k \in \mathbb{N}}$ then we have $$\int_D f \leq \lim_{k \to + \infty} \int_D f_k .$$ This implies that $$\lim_{k \to + \infty} \int_D f_k = \int_D \, f \ .$$
What you did here is a proof of Lebesgue's monotone convergence theorem using Fatou's lemma. Each of them can be derived directly from the definition of Lebesgue integral, you can also show the latter using the former. All these three proofs I've just mentioned can be found on Wikipedia.
And yes, your proof seems all right.