lets-a-and-b-be-two-sequence-such-that-A-lim-n-a-n-B-lim-n-b-n-exist-and-are-finite-lets-c-be-a-new-sequence-c-n-p-n-a-n-q-n-b-n-p-q-N-0-1-p-n-q-n-1-d-N-d-N-d-d-

Question Number 4507 by 123456 last updated on 03/Feb/16

lets a and b be two sequence such that

A = \lim_{n \to + \infty} a_{n}

B = \lim_{n \to + \infty} b_{n}

exist and are finite, lets

c be a new sequence

c_{n} = p (n) a_{σ (n)} + q (n) b_{μ (n)}

p, q : N \to {0, 1}

p (n) + q (n) = 1

d_{σ} \subset N, d_{μ} \subset N, d_{σ} \cup d_{μ} = N, d_{σ} \cap d_{μ} = \emptyset

ρ : d_{σ} \to N is one to one

τ : d_{μ} \to N is one to one

σ (n) = {\begin{cases} ρ (n) n \in d_{σ} \\ 1 n \notin d_{σ} \end{cases}

μ (n) = {\begin{cases} τ (n) n \in d_{μ} \\ 1 n \notin d_{μ} \end{cases}

proof or give a counter example that

if

A = B

them

\lim_{n \to + \infty} c_{n} exists

Commented by Yozzii last updated on 05/Feb/16

L e t l = \lim_{n \to + \infty} c_{n} = \lim_{n \to + \infty} (p (n) a_{σ (n)} + q (n) b_{μ (n)})

S i n c e p (n) + q (n) = 1 \forall n \in N, \Rightarrow p (n) = 1 - q (n) .

∴ l = \lim_{n \to + \infty} (a_{σ (n)} + q (n) (b_{μ (n)} - a_{σ (n)}))

l = \lim_{n \to + \infty} a_{σ (n)} + (\lim_{n \to + \infty} q (n)) {\lim_{n \to + \infty} (b_{μ (n)} - a_{σ (n)})}

A s n \to + \infty, t h e v a l u e s o f μ (n) a n d σ (n)

a r e b o t h i n d e t e r m i n a t e s i n c e n \in d_{μ} o r

n \in d_{σ} a s n \to + \infty . S o, σ (n) = ρ (n) o r

σ (n) = 1 a s n \to + \infty . S i m i l a r l y, μ (n) = τ (n)

o r μ (n) = 1 a s n \to + \infty . q (n) i s a l s o

i n d e t e r m i n a t e i n v a l u e a s n \to + \infty

s i n c e q (n) = 0 o r q (n) = 1, d e p e n d i n g

o n n .

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