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To solve this problem, we need to work through two parts: ### Part 1: Show that the sequence \(\{f_n(x)\}\), where \[ f_n(x) = \frac{n}{x + n}, \] is uniformly convergent on the interval \([0, k]\), where \(k > 0\). #### Step 1: Pointwise Limit of the Sequence First, find the pointwise limit of the sequence \(\{f_n(x)\}\) as \(n \to \infty\) for a fixed \(x\): \[ \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{n}{x + n}. \] To evaluate the limit: \[ \lim_{n \to \infty} \frac{n}{x + n} = \lim_{n \to \infty} \frac{1}{\frac{x}{n} + 1}. \] As \(n \to \infty\), \(\frac{x}{n} \to 0\), so \[ \lim_{n \to \infty} f_n(x) = \frac{1}{1} = 1. \] Thus, the pointwise limit of the sequence \(\{f_n(x)\}\) is the constant function \(f(x) = 1\). #### Step 2: Uniform Convergence on _____ _______ __________ ____ ________ _________ __________.
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