# Thread: Show that fn + gn uniformly converges to f + g

1. ## 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.

2. ## 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.

3. 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.