Takeover Season
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Hi there,

I know that if a sequence is eventually constant, then it is convergent in any metric space X equipped with the discrete metric. I was wondering if this result holds in general with ANY metric, rather than just the discrete metric?

Thank you.
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Cryptokyo
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(Original post by Takeover Season)
Hi there,

I know that if a sequence is eventually constant, then it is convergent in any metric space X equipped with the discrete metric. I was wondering if this result holds in general with ANY metric, rather than just the discrete metric?

Thank you.
Yes any sequence that is eventually constant in a metric space is convergent to that constant. As for any such sequence  x_{n} we have that we can find an N\in\mathbb{N} such that x_{n} = c \in X for all  n \geq N. So for all  n \geq N,  d(x_{n},c)=0 and thus is convergent to c.

However, a convergent sequence in a discrete metric space must eventually be constant. Whereas, this is not true for any metric space.
Last edited by Cryptokyo; 1 month ago
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Takeover Season
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(Original post by Cryptokyo)
Yes any sequence that is eventually constant in a metric space is convergent to that constant. As for any such sequence  x_{n} we have that we can find an N\in\mathbb{N} such that x_{n} = c \in X for all  n \geq N. So for all  n \geq N,  d(x_{n},c)=0 and thus is convergent to c.

However, a convergent sequence in a discrete metric space must eventually be constant. Whereas, this is not true for any metric space.
Thank you so much, that was actually very beautifully explained.
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