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I am unsure about the following result which I need for a proof.

Let $(a_k) \in \mathbb R$ be a sequence such that $a_k \ge a_{k+1}$ and $\displaystyle \lim_{k\to\infty} a_k = a$ and $a_k > a$ for all $k$. Then the closed set $[a_k, b]$ converges to the half open set $(a, b]$.

My reasoning is that since no interval $[a_k, b]$ contains $a$, their union cannot contain $a$ either. Therefore $$a\notin \lim_{k\to\infty} [a_k,b]$$

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    $\begingroup$ What does it mean for a set to converge to another set? In other words, how is $\lim_{k\to\infty}[a_k, b]$ defined? $\endgroup$ – 5xum Oct 19 '16 at 14:19
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    $\begingroup$ So far, so good. Now you need to give an argument that $(a,b]$ is covered. $\endgroup$ – Daniel Fischer Oct 19 '16 at 14:31

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