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## Homework Statement

[itex]\ell[/itex] is the set of sequences of real numbers where only a finite number of terms is non-zero, and the distance metric is [itex]d(x,y) = sup|x_n - y_n|[/itex], for all n in naural-numbers

then the sequence [tex]u_k = {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}, 0,0,0...}[/tex]

and [itex]\left\{u_k\right\}^{\infty}_{k=1}[/itex] is what i assume to be a sequence of sequences.

I know its a Cauchy sequence. the question ask to show that {u

_{k}} is convergent or not convergent

## The Attempt at a Solution

so here i just see that it will converge to the sequence [itex]\frac{1}{k}[/itex] from k=1 to infinity which converges to 0 but doesn't contain finite non-zero numbers. so the squenec doest belong to the set [itex]\ell[/itex] and its not convergent in [itex]\ell[/itex]

Is this correct or completely wrong. I have a feeling its not and it needs more epsilons. Hard thing with proof is that there is no way to check my answer. if its correct i have some follow-up questions