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I should be able to figure this out, but it has me a bit confused conceptually. I'm really just not sure how to approach it in a rigorous fashion. Any help?

If $a_0, a_1, a_2, . . .$ is a decreasing sequence of positive numbers, then the alternating series $a_0−a_1+a_2−a_3+a_4−a_5+· · ·$ converges to some value $L$. Prove that $0 < L < a_0$.

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Note

  • Since $(a_{n})$'s are decreasing so $a_{1} > a_{2} > a_{3} > \cdots >a_{n}>\cdots$

  • $a_{0}-a_{1} + a_{2} - a_{3} + \cdots = a_{0} -\bigl(a_{1}-a_{2}\bigr) -\bigl(a_{3}-a_{4}\bigr)-\cdots<a_{0}$

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    $\begingroup$ You should also add that $(a_0 - a_1) + (a_2-a_3) + \cdots > 0$ for the other bound. +1 though. $\endgroup$ – Macavity May 30 '14 at 7:04
  • $\begingroup$ @Macavity: Yeah! I thought Jessie could have done that. :D $\endgroup$ – crskhr May 30 '14 at 8:11

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