Sunday, 23 April 2017

real analysis - Proof that limlimitsntoinftysqrtxn=sqrtlimlimitsntoinftyxn

It is asked to prove that limnxn=limnxn, and suggested to use the following two inequalities:



a+ba+2ab+b


bbab



The second inequality holds iff ab0.



I've tried different possibilities, but couldn't figure out how to either take the limit sign out of the square root, or take the limit sign into the square root. Would appreciate some hints, but not an entire solution please.

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