normed spaces - Convergence of sequence of function in norm.

Let $1\leq p<\infty$. Suppose that $\{f_k\}$ is a sequence in $L^p(X,\mathcal{M},\mu)$ such that the limit

$f(x)=\lim_{k \to \infty}f_k(x)$

exists for $\mu$-a.e. $x\in X$. Asumme that

$\liminf_{k \to \infty} \|f_k \|_p=a$ is finite. Then prove that

a) $f \in L^P$

b) $\|f \|_p\leq a$

c) Assuming that $\|f \|_p=\lim_{k\to \infty}\|f_k \|_p$,

$\lim_{k\to \infty} \|f-f_k\|_p=0$

By Fatous' Lemma, I proved b). Is the fact that $f(x)=\lim_{k \to \infty}f_k(x)$ exists $\mu$-a.e. guarantees a)?.

For c), using inequality $(a+b)^p \leq 2^{p-1}(a^p+b^p)$, I hope

$|f-f_k|^p \leq 2^{p-1}(|f|^p+|f_k|^p)$ hold. Then I can use dominated convergence theorem. But with $f\in L^p$, I don't know how to proceed.

Any help will be thankful.