Expression for Set of Points at which Sequence of Functions does not Converge to Given Function

Theorem
Let $X$ be a set.

Let $f : X \to \R$ be a real function.

For each $n \in \N$, let $f_n : X \to \R$ be a real function.

Then we have:


 * $\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty \set {x \in X : \size {\map {f_n} x - \map f x} \ge \frac 1 k}$

Proof
It helps to rewrite:

We have that $\sequence {\map {f_n} x}_{n \in \N}$ does not converge to $\map f x$ :


 * there exists a real number $\epsilon > 0$ such that for each $N \in \N$ there exists $n \ge N$ such that $\size {\map {f_n} x - \map f x} \ge \epsilon$

Clearly then, setting $\epsilon = 1/k$, if $x \in X$ is such that:


 * there exists $k \in \N$ such that for each $N \in \N$ there exists $n \ge N$ such that $\size {\map {f_n} x - \map f x} \ge \dfrac 1 k$

we have that:


 * $\sequence {\map {f_n} x}_{n \in \N}$ does not converge.

So:


 * $\ds \set {x \in X : \text {there exists some } k \in \N \text { such that for each } N \in \N \text { there exists } n \ge N \text { such that } \size {\map {f_n} x - \map f x} \ge \frac 1 k} \subseteq \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x}$

Now suppose that $x \in X$ is such that:


 * there exists a real number $\epsilon > 0$ such that for each $N \in \N$ there exists $n \ge N$ such that $\size {\map {f_n} x - \map f x} \ge \epsilon$

Pick $k \in \N$ sufficiently large so that:


 * $\epsilon \ge \dfrac 1 k$

Then we have that:


 * there exists $k \in \N$ such that for each $N \in \N$ there exists $n \ge N$ such that $\size {\map {f_n} x - \map f x} \ge \dfrac 1 k$.

So:


 * $\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} \subseteq \set {x \in X : \text {there exists some } k \in \N \text { such that for each } N \in \N \text { there exists } n \ge N \text { such that } \size {\map {f_n} x - \map f x} \ge \frac 1 k}$

Then from the definition of set equality:


 * $\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \set {x \in X : \text {there exists some } k \in \N \text { such that for each } N \in \N \text { there exists } n \ge N \text { such that } \size {\map {f_n} x - \map f x} \ge \frac 1 k}$

That is:


 * $\ds \set {x \in X : \sequence {\map {f_n} x}_{n \in \N} \text { does not converge to } \map f x} = \bigcup_{k \mathop = 1}^\infty \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty \set {x \in X : \size {\map {f_n} x - \map f x} \ge \frac 1 k}$