Imum Coeli
Junior Member
- Joined
- Dec 3, 2012
- Messages
- 86
I have another question and I'm not sure if my answer can even be called a proof...
Question: Let \(\displaystyle (a_n)\) be a bounded sequence. Define two new sequences \(\displaystyle (b_k) \) and \(\displaystyle (c_k) \) by
\(\displaystyle b_k := sup\{a_n|n \geq k\}, \; c_k := inf\{a_n|n \geq k\} , \; k = 1,2,3,...\)
a) Show that \(\displaystyle (b_k) \) is decreasing and \(\displaystyle (c_k) \) is increasing
b) Briefly explain why the sequences converge
Answer:
a) Since \(\displaystyle (b_1) \) is taken on the largest possible interval of \(\displaystyle (a_n) \) then by definition \(\displaystyle (b_1) \) is the largest possible value for all \(\displaystyle (a_n) \) therefore \(\displaystyle (b_1) \geq (b_k) \; k>1, \; k \in \mathbb{N} \).
Likewise \(\displaystyle (b_k) \) is always the suprema of \(\displaystyle (a_n) \) taken on a larger interval that \(\displaystyle (b_{k+1}) \) then by the same argument we get \(\displaystyle (b_k) \geq (b_{k+1}) \; \forall k \in \mathbb{N} \)
Hence \(\displaystyle (b_k) \) is decreasing.
Similarly for \(\displaystyle (c_k) \)
b) The sequences converge because they are both bounded and monotone.
Thanks
Question: Let \(\displaystyle (a_n)\) be a bounded sequence. Define two new sequences \(\displaystyle (b_k) \) and \(\displaystyle (c_k) \) by
\(\displaystyle b_k := sup\{a_n|n \geq k\}, \; c_k := inf\{a_n|n \geq k\} , \; k = 1,2,3,...\)
a) Show that \(\displaystyle (b_k) \) is decreasing and \(\displaystyle (c_k) \) is increasing
b) Briefly explain why the sequences converge
Answer:
a) Since \(\displaystyle (b_1) \) is taken on the largest possible interval of \(\displaystyle (a_n) \) then by definition \(\displaystyle (b_1) \) is the largest possible value for all \(\displaystyle (a_n) \) therefore \(\displaystyle (b_1) \geq (b_k) \; k>1, \; k \in \mathbb{N} \).
Likewise \(\displaystyle (b_k) \) is always the suprema of \(\displaystyle (a_n) \) taken on a larger interval that \(\displaystyle (b_{k+1}) \) then by the same argument we get \(\displaystyle (b_k) \geq (b_{k+1}) \; \forall k \in \mathbb{N} \)
Hence \(\displaystyle (b_k) \) is decreasing.
Similarly for \(\displaystyle (c_k) \)
b) The sequences converge because they are both bounded and monotone.
Thanks