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GATE 2022 MA – Question 40

Real Analysis · Sequences and Series of Functions · 2 marks · Multiple choice

On [0,1] let f_n(x)=x²/[x²+(1−nx)²]. Which holds?

  1. The sequence is not equicontinuous.
  2. The sequence converges uniformly.
  3. The sequence is equicontinuous.
  4. It is uniformly bounded and has a uniformly convergent subsequence.

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Show answer and explanation

Correct answer: (A) The sequence is not equicontinuous.

Explanation

For every fixed x, f_n(x)→0, including x=0. However f_n(1/n)=1 while f_n(0)=0, despite 1/n→0. This violates equicontinuity and uniform convergence. Every subsequence still has supremum 1 and pointwise limit zero, so no uniformly convergent subsequence exists. **A**.

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