Correct Answer - Option 3 : unbounded above in
R
Concept:
A natural number is a number that occurs commonly and obviously in nature. As such, it is a non-negative number. The set of natural numbers can be denoted by
N = {1, 2, 3, 4,....}
The set of natural numbers is bounded below and not bounded above in R.
We can prove not bounded above in R using contradiction.
Proof:
Assume by way of contradiction that N is a bounded above. Then, since N is not empty, it follows from the completeness axiom that sup(N) exists. Thus there must be m ∈ N such that
sup(N) - 1 < m (sup(N) means supremum of N or least upper bound)
⇒ sup(N) < m + 1
As m ∈ N , also m + 1 ∈ N, Which is a contradiction.
∴ N is not bounded above