Closure of Infinite Union may not equal Union of Closures/Proof 1

Proof
Proof by Counterexample:

Consider the real number line $\struct {\R, \tau_d}$ with the usual (Euclidean) topology $\tau_d$.

Let:
 * $H_n \subseteq \R: H_n = \closedint {\dfrac 1 n} 1$ for $n \ge 2$

where $\closedint {\dfrac 1 n} 1$ denotes the closed real interval from $\dfrac 1 n$ to $1$.

From Closed Real Interval is Closed Set, $\closedint {\dfrac 1 n} 1$ is a closed set of $\struct {\R, \tau_d}$.

Then from Set is Closed iff Equals Topological Closure:
 * $\map \cl {H_n} = H_n$

Also:
 * $\displaystyle \bigcup_{n \mathop \ge 2} \map \cl {H_n} = \bigcup_{n \mathop \ge 2} H_n = \hointl 0 1$

However:
 * $\displaystyle \map \cl {\bigcup_{n \mathop \ge 2} H_n} = \closedint 0 1$

So:
 * $\displaystyle \bigcup_{n \mathop \ge 2} \map \cl {H_n} \ne \map \cl {\bigcup_{n \mathop \ge 2} H_n}$