= Solution
A <metric space> $(X,d)$ is a <bounded set> if there is a finite constant $M\geq0$ such that
$$
\boxed{d(x,y)\leq M\quad\text{for every }x,y\in X}.
$$
For nonempty $X$, this is equivalent to containment in an <open ball> of finite radius: the <triangle inequality> converts a radius bound into a diameter bound, and conversely a diameter bound gives a ball of any slightly larger radius about a chosen point.
It is a <complete metric space> if every <Cauchy sequence> in $X$ has a <limit of a sequence> belonging to $X$. Explicitly, the hypothesis on $(x_n)$ is
$$
\forall\epsilon>0\ \exists N\ \forall m,n\geq N:\ d(x_m,x_n)<\epsilon,
$$
and <completeness> requires an $x\in X$ with $d(x_n,x)\to0$. Thus
$$
\boxed{\text{every Cauchy sequence converges in }X}.
$$
Boundedness controls distances; <completeness> ensures that these limiting points remain in the space.
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