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Let $(X,d)$ be a metric space. Then it MUST BE TRUE that:

every sequence $(x_n)_n$ in $X$ has a unique limit in $(X,d)$.

a sequence $(x_n)_n$ in $X$ can have at most one limit.

every convergent sequence $(x_n)_n$ in $(X,d)$ is also a Cauchy sequence.

every Cauchy sequence $(x_n)_n$ in $(X,d)$ is also a convergent sequence.

a sequence $(x_n)_n$ in $X$ can have at most one subsequential limit.

every Cauchy $(x_n)_n$ sequence is bounded.