Real Analysis

A set A is said to be countable if there exists a function f: A→N such that

• f is bijective

• f is surjective

• f is identity map

• f is vector

Answer explanation

By definition of countable set, it must be bijective.

Let A={x|x∈N ∧ x²≤7 }⊂N. Then supremum of A is

• 7

• 3

• 2

• does not exist

Answer explanation

In tabular form A={1,2}and set of upper bouds is {2,3,4,...}Now supremum is least upper bound 2.

A convergent sequence has only ……………. limit(s).

• one

• two

• three

• none of these

Answer explanation

limit of the sequence, if it exist, is unique.

If the sequence is convergent then

• it has two limits.

• it is bounded.

• it is bounded above but may not be bounded below.

• it is bounded below but may not be bounded above.

Answer explanation

If a sequence of real numbers is convergent, then it is bounded.

A sequence {(−1)ⁿ} is

• convergent.

• unbounded.

• divergent.

• bounded.

Answer explanation

|(−1)ⁿ|=1<1.1 for all n∈N, therefore it is bounded.

• bounded.

• unbounded.

• divergent.

• none of these

Answer explanation

Definition of Cauchy sequence.

If the sequence is decreasing, then it …………….

• converges to its infimum.

• diverges.

• may converges to its infimum

• is bounded.

Answer explanation

If the sequence is bounded and decreasing, then it convergent.

• the term of the sequence lies in a vertical strip of finite width

Answer explanation

It is a definition of bounded sequence.