2. Prove that a Cauchy sequence is a bounded sequence. Proof: By ...
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2. Prove that a Cauchy sequence is a bounded sequence. Proof: By ...
2. Prove that a Cauchy sequence is a bounded sequence. Proof: By definition,
given any e > 0 there is an integer N such that. |an - am| < e for all n, m ³ N. So.
2. Prove that a Cauchy sequence is a bounded sequence. Proof: By definition, given any e > 0 there is an integer N such that |an - am| < e for all n, m ³ N. So |an| -|am| £ |an - am| < e for all n, m ³ N. Taking
m = N and transposing, we have |an| < |aN| + e for all n ³ N.
Thus, for all n, |an| £ max{|a1|, ...,|aN-1|,|aN| + e }. A abound for {an} is, then, M = max{|a1|, ...,|aN-1|,|aN| + e }.
