Question

16. Which of the following is true?
(A) Every uniformly convergent sequence is pointwise convergent
() Every pointwise convergent sequence is uniformly convergent.
(C) Every sequence has a convergent subsequence.
(D) None of these​

Answer 1

the correct answer is d

Answer 2

(A) Every uniformly convergent sequence is pointwise convergent

Uniform convergence:

A sequence on an interval [a, b] to any function f for any ε > 0 and x ∈ [a, b] there exist integer N which is independent of x but depends on ε so that every x ∈ [a, b]

$|f_{n}(x) - f(x)|$ < ε,        [∀ n ≥ N]

Every uniformly convergent sequence is pointwise convergent, and uniform limit function is similar to the pointwise limit function.

Therefore, every uniformly convergent sequence is pointwise convergent.