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fa.functional analysis – Understanding a proof about the limit of a suite of open games

We read a proof about the following limit
begin {equation} tag {1}
lim_ {n to infty} sigma_1 (T_n) = sigma_1 (T),
end {equation}

or $ T: D (T) subseteq H to H $ and $ T_n: D (T_n) subseteq H to H $ are linear operators on a Hilbert space $ H $ and
$$ sigma_1 (A) = sigma (A) cup {z in mathbb {C}: | (z-A) ^ {- 1} |> 1 } $$
for a linear operator $ A $. In the evidence, it is said that it is sufficient to show the following:

(i) If $ K subseteq sigma_1 (T) $ is compact then there is $ N in mathbb {N} $ such as $ K subseteq sigma_1 (T_n) $ for everyone $ n geq N $.

(ii) if K $ is compact and $ K cap overline { sigma_1 (T)} = emptyset $ then there is $ N in mathbb {N} $ such as $ K cap overline { sigma_1 (T_n)} = emptyset $ for everyone $ n geq N $.

But we do not know why (i) and (ii) implies (1)?

The definition of the convergence of the sets used in (1) is as follows:
Let $ {X_n } $ a sequence of subsets of $ mathbb {C} $. then $ x in limsup X_n $ if there is a subsequence $ {X_ {n_k} } $ and a sequence $ {x_k } $ such as $ x_k in X_ {n_k} $ and $ lim_ {k to infty} x_k = x $. On the other hand, $ x in liminf X_n $ if there is a sequence $ {x_n } $ with $ x_n in X_n $ that sucks $ lim_ {n to infty} x_n = x $. The limit $ lim_ {n to infty} X_n $ exists if $ limsup X_n = liminf X_n $ and we define $ lim_ {n to infty} X_n = limsup X_n $.

Our attempt:

We believe that (i) implies that $ sigma_1 (T) subseteq liminf sigma_1 (T_n) $ because if $ z in sigma_1 (T) $ then $ {z } $ is compact.

From (ii) we can get that $ limsup overline { sigma_1 (T_n)} subseteq overline { sigma_1 (T)} $. Indeed if $ z notin overline { sigma_1 (T)} $, then there is $ varepsilon> $ 0 such as $ overline {B (z; varepsilon)} cap overline { sigma_1 (T)} = emptyset $ and (ii) we get $ overline {B (z; varepsilon)} cap overline { sigma_1 (T_n)} = emptyset $ for all big enough $ n $, so $ z notin limsup overline { sigma_1 (T_n)} $.

But how can we show that $ limsup sigma_1 (T_n) subseteq sigma_1 (T) $?

Thank you for any help you can provide us.