Question

Every bounded sequence has a cluster point; then this theorem is known as:
1. Cauchy's theorem
2. Weierstrass theorem
3. Bolzano-weierstrass theorem
4. None of these

Answer 1

Correct Answer - Option 3 : Bolzano-weierstrass theorem

Concept:

Every bounded infinite set of real numbers has at least one limit point or cluster point. The theorem is known as Bolzano-Weierstrass Theorem

Some other important theorems by Bolzano-Weierstrass are:

1. Every bounded sequence has a convergent subsequence.

2. A sequence {an} converges if and only if it is unbounded and has exactly one subsequential limit

Cauchy's theorem

If f(z) is analytic everywhere within a simply-connected region then:

\(\oint_c f(z)\ dz = 0\)

for every simple closed path C lying in the region