Friday, 9 February 2018

calculus - Showing if limntoinftyan=L then limntoinfty2an=2L using definition




If lim then prove using the limit definition that: \displaystyle \lim_{n\to\infty} -2a_n=-2L.





From the given and the definition we know that: $L-\epsilon-2a_n>-2L-2\epsilon\Rightarrow |-2a_n+2L|<2\epsilon which concludes that: \displaystyle \lim_{n\to\infty} -2a_n=-2L$.



I feel like I was cheating by doing that multiplication by -2, is it alright? it's still true for 2\epsilon right?


Answer



The general idea of you solution is very much correct. The only thing that is missing is that we have: $L-\epsilonfor sufficiently large n.


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