Consider a sequence of integers defined recursively as follows: a) r1 = 2, b) r+1 is divisible by 5", and c) n+1 -n is divisble by 5". 5) Prove that if m

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Consider a sequence of integers defined recursively as follows:
a) ri = 2,
b) , +1 is divisible by 5", and
c) rn+1 - In is divisble by 5".
5) Prove that if m <n then r,-m is divisible by 5".
6) Prove that the sequence is Cauchy.
7) Prove that in (Q, ds) we have that lim (r +1) 0.
Notice that if r, converged to c in (Q, d5), then c+1= 0. This is impossible in Q, so the
sequence doesn't converge. This shows (Q, dz) is not complete.
Transcribed Image Text:Consider a sequence of integers defined recursively as follows: a) ri = 2, b) , +1 is divisible by 5", and c) rn+1 - In is divisble by 5". 5) Prove that if m <n then r,-m is divisible by 5". 6) Prove that the sequence is Cauchy. 7) Prove that in (Q, ds) we have that lim (r +1) 0. Notice that if r, converged to c in (Q, d5), then c+1= 0. This is impossible in Q, so the sequence doesn't converge. This shows (Q, dz) is not complete.
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