1 Let fn(x) = 1+(nx1). (i) Does fn converge on [0, 1]? If so, to what...

 

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1 Let fn(x) = 1+(nx−1)². (i) Does fn converge on [0, 1]? If so, to what function? (ii) Does fʼn converge uniformly? (b) Prove or give counterexample: if fn → f and 9n → g both converge uniformly, then hnfn 9n converges uniformly to h f.g. . = (c) Suppose fn: R → R is surjective for each n, and that fn →f. Is f necessarily surjective? 1 Let fn(x) = 1+(nx−1)². (i) Does fn converge on [0, 1]? If so, to what function? (ii) Does fʼn converge uniformly? (b) Prove or give counterexample: if fn → f and 9n → g both converge uniformly, then hnfn 9n converges uniformly to h f.g. . = (c) Suppose fn: R → R is surjective for each n, and that fn →f. Is f necessarily surjective?