GATE 2025 MA – Question 55
On [0,1], f₁(x)=x/2 and $f_{n+1}=f_n-\frac12(f_n^2-x)$. Which statements hold?
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Correct answer: (B) Converges uniformly; (D) 0≤f_n(x)≤√x for all x,n
Explanation
Initially 0≤f₁≤√x. If 0≤a≤√x≤1, update a+(x−a²)/2 is at least a, and its distance below √x is $(\sqrt x-a)[1-(\sqrt x+a)/2]\ge0$. Thus f_n increases within those bounds, proving D. Its pointwise limit satisfies L²=x, hence L=√x, continuous. Dini’s theorem gives uniform convergence B. C is already false at x=0 because it asserts 0>0. **B,D**.