Thread: Sum of two Uniform Convergent Functional Series

1. Sum of two Uniform Convergent Functional Series

Let $\displaystyle F(x):=\displatstyle\sum\limits_{j=1}^{\infty}f_{j} (x)$ and
$\displaystyle G(x):=\displatstyle\sum\limits_{j=1}^{\infty}g_{j} (x)$

be uniformly convergent functional series.

Show that $\displaystyle \displatstyle\sum\limits_{j=1}^{\infty}(f_{j} + g_{j})$ is uniformly converget to $\displaystyle F + G$

Now I suppose I need to use the Chebyshev Norm and show that $\displaystyle \bigg\lVert\displatstyle\sum\limits_{j=1}^{\infty} (f_{j} + g_{j}) - (F + G)\bigg\rVert_{\infty} <\epsilon$

Or any other method perhaps??
I have no idea other than that one, any help would be much appriciated thanks

2. You only need the standard argument:

$\displaystyle \left |{\displaystyle\sum_{j=1}^n{(f_j(x)+g_j(x))-(F(x)+G(x))}}\right |\leq$

$\displaystyle \left |{\displaystyle\sum_{j=1}^n{f_j(x)-F(x))}}\right |+\left |{\displaystyle\sum_{j=1}^n{(g_j(x)-G(x))}}\right |< \dfrac{ \epsilon}{2}+\dfrac{ \epsilon}{2}= \epsilon$

for every

$\displaystyle x\in D(F)\cap D(G)$

and for

$\displaystyle n\geq n_o$