We have seen the completeness axiom for the reals in the form of the
Monotone Convergence Theorem. This web page discusses an application
of this for bounded sequences and proves a beautiful result: that every
bounded sequence has a convergent subsequence, the
Bolzano-Weierstrass theorem

.

We have seen that every convergent sequence is bounded, but that
not every bounded sequence is convergent. For example, (-1)^{
}
defines a non-convergent bounded sequence. Thus there is no full converse
to the statement that every convergent sequence is bounded.
The Bolzano-Weierstrass theorem, to be presented next, gives a partial converse.

Bolzano-Weierstrass Theorem.

**Proof.**

Suppose that (_{
)
} is bounded. For the purposes of this proof, say that
the _{
} of the sequence is dominant
if _{
}_{
}
for all

Case 1. Suppose that (_{
)
} has infinitely many
dominant terms

where _{1
2
3
}
dominant

we have
_{
1
}_{
2
}_{
3
}
and this gives a bounded monotonic nonincreasing subsequence which
converges by the monotone convergence theorem.

Case 2. If not, then the sequence (_{
)
} has only finitely many
dominant terms. Choose _{1}
_{
}, and define
_{1=+1
}
_{
1
} is not dominant there
is _{2
1
}
_{
1
}_{
2
}
,
and as _{
2
} is not dominant there
is _{3
2
}
_{
2
}_{
3
}
,
and so on. Continuing in this way we obtain a monotonic bounded increasing subsequence
_{
1
}_{
2
}_{
3
}
which once again converges by the monotone convergence theorem.

The Bolzano-Weierstrass theorem is an important and powerful result
related to the so-called compactness of intervals **
**