limit of a sum
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From: lucabenzo on 18 Jul 2010 04:53 Hi everybody, I hope someone can help me with this... Let a_k, b_k \in \mathbb{R}^n. I know: 1) \lim_{k \rightarrow \infty}(a_k) exists 2)\lim_{k \rightarrow \infty}(a_k + b_k) exists. Can I conclude that \lim_{k \rightarrow \infty}(b_k) exist? If it is true, how can I prove this? Thank you
From: Tonico on 18 Jul 2010 05:14
On Jul 18, 11:53 am, lucabenzo <neblina.l...(a)libero.it> wrote: > Hi everybody, I hope someone can help me with this... > > Let a_k, b_k \in \mathbb{R}^n. > > I know: > > 1) \lim_{k \rightarrow \infty}(a_k) exists > > 2)\lim_{k \rightarrow \infty}(a_k + b_k) exists. > > Can I conclude that \lim_{k \rightarrow \infty}(b_k) exist? If it is > true, how can I prove this? > > Thank you b_k = (a_k + b_k) - a_k , difference of converging sequences ==> it converges and by arithmetic of limits its limit is the difference of the limits. Claim: if A = lim a_k, B = lim b_k exist, then exist both lim (a_k + b_k) and lim (a_k - b_k) , and their limit is A + B, A - B resp. Proof: let e > 0 ==> there exist N_1, N_2 in N s.t. 1) for all n > N_1 , |a_n - A| < e/2 2) for all n > N_2 , |b_n - B| < e/2 Take M > max (N_1, N_2)==> for all n > M: |a_n + b_n - (A + B)| <= |a_n - A| + |b_n - B| < e. Tonio Tonio |