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Proving that $\limsup {x_{n}} = \lim_{n \rightarrow \infty} ( \sup _ {m \geq n} x_{m} )$.

I want to prove it using only the definition of the $\limsup {x_{n}}$: that $\limsup {x_{n}}$ is the largest accumulation point of $\{x_{n}\}$.

I have seen many links for the proof here but as far as I understand not directly from this definition and after reading them all I got confused, could any one help me in proving this?

Adrian Keister
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Intuition
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    So do you want to prove that $\limsup {x_{n}} = \lim_{n \rightarrow \infty} ( \sup _ {m \geq n} x_{m} )$ using some other definition, or do you want to use this as a definition, and prove that $\limsup {x_{n}} $ is the largest accumulation point of ${x_n}$? – Aweygan Sep 27 '18 at 22:49
  • I have read this question and I did not understand the solution @Holo and I do not feel that they are the same – Intuition Sep 27 '18 at 22:58
  • @hopefully so please add what didn't you understand to the question, and why do you think it is different – ℋolo Sep 27 '18 at 23:01
  • @Holo for example in my question I did not mention at all equation (1) in you suggested link – Intuition Sep 27 '18 at 23:44

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