A decreasing sequence of integrable functions with the limit of the integral exist

Question

Question: Prove or disprove: If {${f_{n}}$} is a decreasing sequence of integrable functions such that $lim\int f_{n}$ exists in $\Bbb R$, then $lim\int f_{n}=\int lim\ f_{n}$.

Attempt: I think this is right. Since $lim\int f_{n}$ exists in $\Bbb R$, then each function is finite a.e.. Then we can use LDCT to with dominated function be max{|$f_1|,|lim\ f_n|$}. Is this proof right?