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Jamin2112
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Prove that lim ...
Suppose that {an} and {bn} are sequences of real numbers with lim n-->∞ an = A and lim n-->∞ bn = B. Just using the definition of convergence, prove that
lim n-->∞ (an + bn) = A + B.
We say a sequence sn converges to S if there exists an N such that N≤n implies |S-sn|<∂ for all ∂>0.
We know there exists an N such that N≤n implies |A - an|<∂ for all ∂>0; likes with some some N' and bn. Let ∂ instead be ∂/2, and let N''=max(N,N'). For all n≥N'', we have |(A-an)+(B-bn)|≤|A-an|+|B-bn|< ∂/2 + ∂/2 = ∂.
Therefore lim n-->∞ (an+bn)=A+B
Homework Statement
Suppose that {an} and {bn} are sequences of real numbers with lim n-->∞ an = A and lim n-->∞ bn = B. Just using the definition of convergence, prove that
lim n-->∞ (an + bn) = A + B.
Homework Equations
We say a sequence sn converges to S if there exists an N such that N≤n implies |S-sn|<∂ for all ∂>0.
The Attempt at a Solution
We know there exists an N such that N≤n implies |A - an|<∂ for all ∂>0; likes with some some N' and bn. Let ∂ instead be ∂/2, and let N''=max(N,N'). For all n≥N'', we have |(A-an)+(B-bn)|≤|A-an|+|B-bn|< ∂/2 + ∂/2 = ∂.
Therefore lim n-->∞ (an+bn)=A+B