Sequence converges

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  • #1
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1. Homework Statement [/b]
Let Jn : n [tex]\in[/tex]N be a sequence of intervals Jn=[tex]\left[[/tex]an,bn[tex]\right][/tex] such that J1[tex]\supset[/tex]J2[tex]\supset[/tex]...[tex]\supset[/tex]Jn[tex]\supset[/tex]Jn+1[tex]\supset[/tex]...
suppose also that the sequence xn=an-bn converges to 0 as n tends to infinite.Show that there is exactly one point a such that a[tex]\in[/tex]Jn for all n [tex]\in[/tex]N


Homework Equations





The Attempt at a Solution


i don't know how to start it , any clue??
 

Answers and Replies

  • #2
jgens
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Well, I'm dreadfully awful at sequence questions, so take my feedback with a grain of salt. Since the sequence [itex]x_n = b_n - a_n[/itex] converges to zero for arbitrarily large [itex]n[/itex], this means that [itex]\mathrm{inf}(b_n) = \mathrm{sup}(a_n) = c[/itex]. Can you prove that this number [itex]c[/itex] must always be an element of [itex][a_n,b_n][/itex].
 
  • #3
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Well, I'm dreadfully awful at sequence questions, so take my feedback with a grain of salt. Since the sequence [itex]x_n = b_n - a_n[/itex] converges to zero for arbitrarily large [itex]n[/itex], this means that [itex]\mathrm{inf}(b_n) = \mathrm{sup}(a_n) = c[/itex]. Can you prove that this number [itex]c[/itex] must always be an element of [itex][a_n,b_n][/itex].

still confusing
 
  • #4
jgens
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The number [itex]c[/itex] would have the property that [itex]a_n \leq c \leq b_n[/itex] for all natural numbers [itex]n[/itex]. What does this tell you about [itex]c[/itex] and its relationship to the interval [itex][a_n,b_n][/itex]?

Again, I'm awful at these types of proofs, so if another member says something otherwise, I would follow their feedback (I'm just trying making sure that you actually have some feedback).
 

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