11. See #18 of this link
Solution
13. Since $ f_n$ converges uniformly to $f$ , then
$ \exists N $ such that $ \mid f_N(x)-f(x) \mid < 1 $ for all $ x \in A $
And since $f_n$ is bounded on A,say ,by M.
It follows that $ \mid f(x) \mid \leq \,\, \mid f(x)-f_N(x) \mid + \mid f_N(x) \mid < 1+M \quad \forall x \in A $
NOTE: เว้นวรรค คำว่า "ขอบเขต" กับคำว่า "บน A" ในโจทย์ด้วยก็ดีนะครับ
14. Use Weierstrass M-Test
15. Let $S_k$ be partial sum of $ \sum f_n $
For $c = 0$ , Obvious !
For $ c \neq 0 $
Since $ \sum f_n $ converges uniformly to $ f $ then
$ \forall \epsilon >0 \,\, \exists N $ such that $ \mid S_k(x)-f(x) \mid <\frac{\epsilon}{\mid c\mid} \quad (k \geq N , \forall x \in A)$
After this, it's not difficult to follow.
Edit : Add Link of Question 11.