Solution:
Since fn→f uniformly on S and each fn is continuous on S, for given ε>0
there is a δ>0 such that as ∣y−x∣<δ , where y∈S , we have
∣f(y)−f(x)∣<2ε
For this δ>0,∃ a positive integer N1 such that as n≥N1 , we have
∣xn−x∣<δ
Hence, as n≥N1 , we have
∣f(xn)−f(x)∣<2ε ...(1)
Also, since fn→f uniformly on S, given ε>0,∃ a positive integer N≥N1 such that as n≥N , we have
∣fn(x)−f(x)∣<2ε for all x∈S
⇒∣fn(xn)−f(xn)∣<2ε ...(2)
By (1) and (2), we obtain that given ε>0,∃ a positive integer N such that as n≥N , we have
∣fn(xn)−f(x)∣=∣fn(xn)−f(xn)∣+∣f(xn)−f(x)∣<2ε+2ε=ε
⇒fn(xn)→f(x)
Hence, limit function f is also continuous at any point c.
Hence, proved.
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