Let $(X, M, \mu)$ be a measure space, and $f_n:\Omega \rightarrow \mathbb R$ sequence of measurable functions.

How can I show that the set of $x$ that for them $f_n(x)$ has a subsequence that converges to $0$ is a measurable set?

Let $(X, M, \mu)$ be a measure space, and $f_n:X \rightarrow \mathbb R$ sequence of measurable functions.

How can I show that the set of $x$ that for them $f_n(x)$ has a subsequence that converges to $0$ is a measurable set?

Let $$(X, M, \mu)$$ be a measure space.

and $$f_n:\Omega -> R$$ sequence of measurable functions.

how can I show that the set of x that for them $$f_n(x)$$ has a subsequence that converges to 0 is a measurable set?

Let $(X, M, \mu)$ be a measure space, and $f_n:\Omega \rightarrow \mathbb R$ sequence of measurable functions.

How can I show that the set of $x$ that for them $f_n(x)$ has a subsequence that converges to $0$ is a measurable set?