I have to prove that a constant sequence and an eventually cinstant sequence is always convergent.
I tried to do this as follows :
I considered (xn) in X to be a constant sequence such that (xn) = x,x,x.... for all n in N.
Then i considered an open ball centered at x and with radius 1/2 i.e. b[x;1/2).
Since this ball will definitely contain a tail of the sequence (xn) therefore (xn) is convergent.
Is this argument correct.Any arbitrary value of "r" should work, right ?
Also can the convergence of eventually constant sequences be proved on the same lines? How do i decide what value of "r" should be chosen for this case ?