Limit change of double sequence of functions which diverges

Let (X,μ)(X,\mu) be a finite measure space and {gm,n(x)}n,m≥1\{g_{m,n}(x)\}_{n,m\geq 1} be a double sequence of (nonnegative) real valued measurable functions on XX. Suppose that this sequence satisfies
gm,n(x)≤gm+1,n(x)∀x∈X g_{m,n}(x)\leq g_{m+1, n}(x)\,\,\forall x\in X
limn→∞gm,n(x)=cmfor a.e. x∈X\lim_{n\to\infty} g_{m,n}(x)=c_{m}\,\,\text{for a.e. }x\in X
limm→∞cm=∞\lim_{m\to\infty} c_{m}=\infty
limn→∞gm,n(x)=fn(x)∀x∈X\lim_{n\to\infty} g_{m,n}(x)=f_{n}(x)\,\,\forall x\in X
for some real sequence {cm}m≥1\{c_{m}\}_{m\geq 1} and sequence of functions {fn(x)}n≥1\{f_{n}(x)\}_{n\geq 1}. Then
limn→∞fn(x)=∞\lim_{n\to \infty} f_{n}(x)=\infty for a.e. x∈Xx\in X.

Is this true? If not, what kind of conditions can be added to make the proposition true? Thanks in advance.

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