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Math Help - Show that fn + gn uniformly converges to f + g

  1. #1
    Member thaopanda's Avatar
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    Show that fn + gn uniformly converges to f + g

    Let f_{n}, g_{n} : R \rightarrow R for n \in N be functions such that f_{n} \rightarrow f and g_{n} \rightarrow g as n  \rightarrow \infty uniformly on the set E \subset R, where f, g : R \rightarrow R are functions. Show that f_{n} + g_{n} \rightarrow f + g as n  \rightarrow \infty uniformly on the set E.
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  2. #2
    Member thaopanda's Avatar
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    Monotonically increasing

    For each n \in N let f_{n} : [a,b] \rightarrow R where a,b \in R are such that a < b. Show that if F : [a,b] \rightarrow R is such that F_{n} \rightarrow F as n \rightarrow \infty in a pointwise fashion for each n \in N the function F_{n} is monotonically increasing, then F is monotonically increasing.
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  3. #3
    Super Member Deadstar's Avatar
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    limit of f_n = f
    limit of g_n = g

    Fix \epsilon > 0, There exist positive integers n_1, n_2 such that for all n \geq n_1, |f_n - f| < \frac{\epsilon}{2} and for all n \geq n_2, |g_n - g| < \frac{\epsilon}{2}. Define n_0 = \textrm{max}(n_1, n_2). Then for all n \geq n_0,

    |(f_n + g_n) - (f + g)| \leq |f_n - f| + |g_n - g| \leq \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon which proves it.
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