I have drawn on a piece of paper a bunch of different sequences in an $\ell_p$ space that converge to a constant sequence. Each of the elements in this sequence in an $\ell_p$ space are eventually constant.
There are two sequences that are bothering me. The first is the sequence that is always constant, but then 0 at ever $2^n$ term. So
$x_1=c,x_2=c,x_3=c,x_4=0,...,x_8=0,...x_{16}=0,...$
In this fashion. This sequence seems to somehow converge to c even though I can always find an N such that $x_N=0$ and so $x_N$ would lie outside of any epsilon band around c. Why is it that I can say this sequence converges to c? Would it still converge if it were every $2n$ term that was 0?
The other sequence that bothers me is the sequence that is always 0 but then "spikes" at the $2^n$ term to c. So
$x_1=0,x_2=0,x_3=0,x_4=c,x_5=0,...,x_8=c,...,x_{16}=c,....$
This sequence also bothers me. If I want to claim that a sequence in $\ell_p$ converges to c and I find this element in my sequence, does my sequence still converge to c?
These examples seem to come up a lot and they always throw me off because I never think of them when I need to. Why are they so important? Why do they come up in proofs and as counter examples. I even see them C[0,1] as functions that peak at certain values of x and are otherwise 0.