# Thread: Real Analysis - Sequences #3

1. ## Real Analysis - Sequences #3

This question has three individual questions to it, so I'll post them as b,c and d. The instructions say to decide which of the following conjectures are true and which are false. Supply a proof for those that are valid and a counterexample for each that is not valid. Here they are:

b. If fn --> f uniformly on A and g is a bounded function on A, then fn*g --> fg uniformly on A.

c. If fn --> f uniformly on a set A, and if each fn is bounded on A, then f must also be bounded.

d. If fn --> f uniformly on a set A, and if fn --> f uniformly on a set B, then fn --> f uniformly on A U B.

2. Originally Posted by ajj86
This question has three individual questions to it, so I'll post them as b,c and d. The instructions say to decide which of the following conjectures are true and which are false. Supply a proof for those that are valid and a counterexample for each that is not valid. Here they are:

b. If fn --> f uniformly on A and g is a bounded function on A, then fn*g --> fg uniformly on A.

c. If fn --> f uniformly on a set A, and if each fn is bounded on A, then f must also be bounded.

d. If fn --> f uniformly on a set A, and if fn --> f uniformly on a set B, then fn --> f uniformly on A U B.
Note: You may want to wait for confirmation on my solutions/suggestions by more seniro members

Which are you having trouble with? I dont have super much time but Ill try one'

b. True, Im assuming that $\displaystyle f_n$ is totally bounded. So let $\displaystyle |f_n(x)|\leqslant M$, so choose $\displaystyle N$ such that $\displaystyle N>n \implies |f_n(x)-f(x)|<1$ or $\displaystyle f_n(x)-1<f(x)<1+f_n(x)$ this in turn implies $\displaystyle -(M+1)<f(x)<M+1$