Sequence of a bounded variation
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Sequence of a bounded variation
Let \( x_n \) be a sequence in the metric space \( (X, d) \) . We define \( x_n \) to be of a bounded variation if:
$$\sum_{n=1}^{\infty}d\left ( x_n, x_{n+1} \right )<+\infty$$
Prove the following:
a) If \( x_n \) is of a bounded variation then it is a standard / basic sequence. (therefore bounded). Does the converse hold?
b) If \( x_n \) is a standard/ basic sequence , then there exists a subsequence of a bounded variation.
c) If every subsequence of \( x_n \) is of a bounded variation , then \( x_n \) is a basic/ standard sequence.
$$\sum_{n=1}^{\infty}d\left ( x_n, x_{n+1} \right )<+\infty$$
Prove the following:
a) If \( x_n \) is of a bounded variation then it is a standard / basic sequence. (therefore bounded). Does the converse hold?
b) If \( x_n \) is a standard/ basic sequence , then there exists a subsequence of a bounded variation.
c) If every subsequence of \( x_n \) is of a bounded variation , then \( x_n \) is a basic/ standard sequence.
Imagination is much more important than knowledge.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 17 guests