Question 2.39: Given limn→∞ an = A and limn→∞ bn = B. Prove that limn→∞ (an...
Given \lim _{n \rightarrow \infty} a_n=A \text { and } \lim _{n \rightarrow \infty} b_n=B . Prove that \lim _{n \rightarrow \infty}\left(a_n+b_n\right)=A+B .
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By definition, given \epsilon we can find N such that
\left|a_n-A\right|<\epsilon / 2,\left|b_n-B\right|<\epsilon / 2 \text { for } n>NThen for n>N,
\left|\left(a_n+b_n\right)-(A+B)\right|=\left|\left(a_n-A\right)+\left(b_n-B\right)\right| \leq\left|a_n-A\right|+\left|b_n-B\right|<\epsilonwhich proves the result.
It is seen that this parallels the proof for limits of functions [Problem 2.28].
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