# Proving Non-Convergence of (X(n)) to X in l∞(R)

• gtfitzpatrick
In summary, we consider the sequence space l^infinity(R) of bounded real sequences with the supremum norm. For a sequence (X(n)) in l^infinity(R) and a sequence X in l^infinity(R), (X(n)) converges to X means that the supremum norm of the difference between (X(n)) and X approaches 0 as n approaches infinity. In this problem, we are given two sequences (X(n)) = (1,1,...,1,0,0,...) and X = (1,1,1,...). We need to prove that (X(n)) does not converge to X. To do this, we can use the definition of the l^infinity norm and show that
gtfitzpatrick

## Homework Statement

consider the sequence space l$$\infty$$(R) of bounded real sequences with the sup norm. If (X(n)) is sequence in l$$\infty$$(R) and X $$\in$$ l$$\infty$$(R), what does it mean to say that (X(n)) converges to X

let (X(n)) be the sequence (1,1,---,1,0,0,---) with the first n coordinates 1 and the rest 0. And let x be the sequence (1) with every coordinate 1. Prove that the sequence (X(n)) does not converge to x

## The Attempt at a Solution

not sure where to start with this. any points to where i can look up info or where to start please?

Look up the definition of the l^infinity norm. If {an} and {bn} are sequences. ||{an}-{bn}||_infinity is the sup of |an-bn| over all n.

## 1. How do you define convergence in l∞(R)?

In l∞(R), convergence is defined as the sequence (X(n)) approaching a limit X, where for every ε>0, there exists an N such that for all n>N, ||X(n)-X||<ε.

## 2. What does it mean for (X(n)) to not converge to X in l∞(R)?

If (X(n)) does not converge to X in l∞(R), it means that the sequence does not approach a limit in the space l∞(R). This could happen if the sequence is unbounded or if it oscillates between values without approaching a specific value.

## 3. How do you prove non-convergence of (X(n)) to X in l∞(R)?

To prove non-convergence, one needs to show that for any limit X, there exists an ε>0 such that for all N, there exists an n>N where ||X(n)-X||≥ε. This means that no matter how large N becomes, there will always be terms in the sequence that are farther away from X than the chosen ε.

## 4. Can (X(n)) still converge to X in a different metric space?

Yes, (X(n)) can still converge to X in a different metric space. Convergence is specific to the metric space being used, and different metric spaces may have different definitions of convergence. Therefore, a sequence that does not converge in one metric space may still converge in another.

## 5. Is non-convergence of (X(n)) to X in l∞(R) the same as divergence?

No, non-convergence and divergence are not the same. Non-convergence refers to the inability of a sequence to approach a specific limit in a given metric space. Divergence, on the other hand, means that the terms in the sequence are becoming infinitely large or infinitely small, and the sequence does not have a limit.

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