Proof for convergent series that there is a B s.th. lxn​l > B for all n

[srosen]

I have attached the pages of the question and the solution. Can someone explain to me what is going on here please? Thank you.

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\(\displaystyle \mbox{12.7 Suppose that }\, \left\{x_n\right\}_{n=1}^{\infty}\, \mbox{ is a sequence of real numbers}\)

. . ..\(\displaystyle \mbox{that converges to }\, x_0\, \mbox{ and that all }\, x_n\, \mbox{ and }\, x_0\, \mbox{ are nonzero.}\)

. . ..\(\displaystyle \mbox{a) Prove that there is a positive number }\, B\, \mbox{ such that }\,\)

. . . . .\(\displaystyle \large{|}\)\(\displaystyle \, x_n \, \large{|}\)\(\displaystyle \, \geq\, B\, \mbox{for all }\, n.\)

. . ..\(\displaystyle \mbox{b) Using (a), prove that }\, \left\{1\, /\,x_n\right\}\, \mbox{ converges to }\, \left\{1\,/\,x_0\right\}.\)

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\(\displaystyle \mbox{12.7 (a) There exists }\, N\, >\, 0\, \mbox{ such that }\, \)

. . . . . . . . . .\(\displaystyle n\, \geq\, N\, \Longrightarrow\, \large{|}\)\(\displaystyle \, x_n\, -\, x_0\, \large{|}\)\(\displaystyle \, <\, \dfrac{1}{2}\, \large{|}\)\(\displaystyle \, x_0\large{|}\)

. . . . . .\(\displaystyle \mbox{Then }\)

. . . . . . . . . .\(\displaystyle \large{|}\)\(\displaystyle \, x_0\, \large{|}\)\(\displaystyle \, -\, \large{|}\)\(\displaystyle \, x_n\, \large{|}\)\(\displaystyle \, \leq\, \large{|}\)\(\displaystyle \, x_0\, -\, x_n\, \large{|}\)\(\displaystyle \, <\, \dfrac{1}{2}\, \large{|}\)\(\displaystyle \, x_0\, \large{|}\)

. . . . . .\(\displaystyle \mbox{or }\)

. . . . . . . . . .\(\displaystyle \dfrac{1}{2}\, \large{|}\)\(\displaystyle \, x_0\, \large{|}\)\(\displaystyle \, \leq\, \large{|}\)\(\displaystyle \, x_n\, \large{|}\)

. . . . . .\(\displaystyle \mbox{for all }\, n\, \geq\, N.\)

. . . . . .\(\displaystyle \mbox{Let }\, B\, =\, \mbox{min}\large\{\)\(\displaystyle \, \dfrac{1}{2}\, \large{|}\)\(\displaystyle \,x_0\, \large{|},\)\(\displaystyle \,\large{|}\)\(\displaystyle \,x_1\, \large{|},\)\(\displaystyle \,\large{|}\)\(\displaystyle \,x_2\, \large{|},\)\(\displaystyle \,...\, \large{|}\)\(\displaystyle \,x_N\, \large{|}\)\(\displaystyle \,\large\}.\)

. . . . . .\(\displaystyle \mbox{For }\, n\, \leq\, N,\, \large{|}\)\(\displaystyle \,x_n\, \large{|}\)\(\displaystyle \,\geq\, \mbox{min}\large\{\)\(\displaystyle \, \large{|}\)\(\displaystyle \,x_j\, \large{|}\)\(\displaystyle \,:\, 1\, \leq\, j\, \leq\, N\, \large\}\)\(\displaystyle \, \geq\, B.\)

. . . . . .\(\displaystyle \mbox{For }\, n\, \geq\, N,\, \large{|}\)\(\displaystyle \,x_n\, \large{|}\)\(\displaystyle \,\geq\, \dfrac{1}{2}\large{|}\)\(\displaystyle \,x_0\, \large{|}\)\(\displaystyle \,\geq\, B.\)

\(\displaystyle \mbox{12.7 (b) Let }\, B\, \mbox{ be as in part (a). Let }\, \epsilon\, >\, 0.\, \mbox{ Choose }\, N\, \mbox{ such that, for }\)

. . . . . .\(\displaystyle n\, \geq\, N,\, \large{|}\)\(\displaystyle \,x_n\, -\, x_0\, \large{|}\)\(\displaystyle \,\leq\, \epsilon\, B\, \large{|}\)\(\displaystyle \,x_0\, \large{|}\)\(\displaystyle \, \mbox{ Then, for }\, n\, \geq\, N,\)

. . . . . . . . . . . .\(\displaystyle \bigg|\,\dfrac{1}{x_n}\, -\, \dfrac{1}{x_0}\, \bigg{|}\)\(\displaystyle \, =\, \dfrac{|\, x_n\,-\, x_0\, |}{|\,x_n\,|\, |\,x_0\,|}\, \leq\, \dfrac{|\,x_n\, -\, x_0\,|}{B\, |\,x_0\,|}\)

. . . . . .\(\displaystyle \mbox{(since }\, |\,x_n\, |\,\geq\, B\, \mbox{ for all }\,n\, \Longrightarrow\, 1\, /\, |\,x_n\, |\,\leq\, 1/B\, \mbox{ for all }\, n)\)

. . . . . . . . . . . .\(\displaystyle \leq\, \dfrac{\epsilon\, \cdot\, B\, |\, x_0\,|}{B\, |\, x_0\,|}\, =\, \epsilon\)

. . . . . .\(\displaystyle \mbox{Therefore, }\, 1\, /\, x_n\, \rightarrow\, 1\, /\, x_0.\)

 

[Ishuda]

I have attached the pages of the question and the solution. Can someone explain to me what is going on here please? Thank you.

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