This seems trivial but I just want to be careful about it. By definition, I would like lim n → ∞ P ( | X n − μ | &gt; ϵ ) = 0 for ϵ &gt; 0. But can I just rewrite it as lim n → ∞ P ( | X n − lim n → ∞ μ n | &gt; ϵ ) = 0 and bring the limit out from there? That seems really odd to me.

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This seems trivial but I just want to be careful about it. By definition, I would like       lim          n      →      ∞        P  (      |        X    n    −  μ      |    &amp;gt;  ϵ  )  =  0 for   ϵ  &amp;gt;  0. But can I just rewrite it as       lim          n      →      ∞        P  (      |        X    n    −      lim          n      →      ∞            μ    n        |    &amp;gt;  ϵ  )  =  0 and bring the limit out from there? That seems really odd to me.

Leland Ochoa · Accepted Answer

There is a problem with writing      lim          n      →      ∞        P  (      |        X    n    −      lim          n      →      ∞            μ    n        |    &amp;gt;  ϵ  )  =  0which is that the n in the outer limit and the n in the inner limit are two different dummy variables. Each is only defined with the confines of its own limit. But because one limit is inside the other the n in       μ    n   is ambiguous. Is it the n of the inner limit, or of the outer limit? In this case, it isn&#039;t hard to figure out, but you shouldn&#039;t have to figure it out.The correct usage is to have different variable names:      lim          n      →      ∞        P  (      |        X    n    −      lim          m      →      ∞            μ    m        |    &amp;gt;  ϵ  )  =  0I&#039;m sure that spoils your next step, which I guess would have been to say      lim          n      →      ∞        P  (      |        X    n    −      μ    n        |    &amp;gt;  ϵ  )  =  0But this is something you need to justify in a more rigorous fashion.

This seems trivial but I just want to be careful about it. By definition, I would like <munder>

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