Suppose that \(\{f_n\}\) is a sequence of functions defined by : \(f_n(x) = \frac{x^n}{1 + x^n}, \ 0 \le x \le 1.\) Then, which of the following statements is true ? \(S_1 : f_n \to f \equiv 0\) uniformly on \([0, 1]\) \(S_2 : \sup_{0 \le x \le \frac{1}{2}} f_n(x) \to F(x) \equiv 0\) on \([0, \frac{1}{2}]\) \(S_3 : f_n \to f = 0\) uniformly on \([0, \frac{1}{2}]\)
Only \(S_2\) and \(S_3\) are true
For \(0 \le x < 1\): \(x^n \to 0 \Rightarrow f_n(x) \to 0\); at \(x = 1\): \(f_n(1) = \frac{1}{2}\) for all \(n\).
So the pointwise limit on \([0, 1]\) is not \(\equiv 0\) (it equals \(\frac{1}{2}\) at \(x = 1\)); \(S_1\) is false.
\(f_n\) is increasing in \(x\) \(\Rightarrow \sup_{0 \le x \le \frac{1}{2}} f_n(x) = f_n(\frac{1}{2}) = \frac{(1/2)^n}{1 + (1/2)^n} \to 0\); \(S_2\) is true.
\(\sup_{[0, 1/2]} |f_n(x) - 0| \le \left(\frac{1}{2}\right)^n \to 0 \Rightarrow f_n \to 0\) uniformly on \([0, \frac{1}{2}]\); \(S_3\) is true.
Hence, Only \(S_2\) and \(S_3\) are true.
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