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I need some fuuther help in order to fully understand Example 4.1.1 ...

Example 4.1.1 reads as follows:

In the above example from Singh we read the following:

" ... ...Then the complement of \(\displaystyle \{ x_n \ | \ x_n \neq x \text{ and } n = 1,2, ... \}\) is a nbd of \(\displaystyle x\). Accordingly, there exists an integer \(\displaystyle n_0\) such that \(\displaystyle x_n = x\) for all \(\displaystyle n \geq n_0\). ... ... "

My question is as follows:

Why, if the complement of \(\displaystyle \{ x_n \ | \ x_n \neq x\) and \(\displaystyle n = 1,2, ... \}\) is a nbd of \(\displaystyle x\) does there exist an integer \(\displaystyle n_0\) such that \(\displaystyle x_n = x\) for all \(\displaystyle n \geq n_0\). ... ... ?

Help will be much appreciated ... ...

Peter

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It may help readers of the above post to have access to Singh's definition of a neighborhood and to the start of Chapter 4 (which gives the relevant definitions) ... so I am providing the text as follows:

Hope that helps ...

Peter