showing that the limit exists

hi i was wondering if anyone could help me with this problem to do with limits. its a real anaylsis subject.

if **g**, defined on an interval **[0,infinity)**, is decreasing function and is bounded below

then i want to show that **lim **x-> infinity **g(x)** exists.

can anyone help? thanks

can someone pls check this solution

hi i have another question, which i have attempted fully but i was wondering if someone could check it is right.

QUESTION:

Suppose that **lim x-> c f(x) = L** where L >0, and that **lim x-> c g(x) = infinity**.

I want to show that lim x->c f(x)g(x) = infinity

MY SOLUTION

Using properties of limits theorem

let f,g be real functions , **lim x-> c f(x)g(x) = L x infinity**

where L x infinity = infinity

hence **lim x-> c f(x)g(x) = inifinity**

as required.

can someone please check this thanks