# Prove that Lim x->c f(x)g(x)=0 if Lim x->c f(x)=0 and |g(x)|<K

Homework Statement:
Suppose Lim x->c f(x)=0 and that g(x) is bounded by a number K>0 such that |g(x)|<K. Define h by h(x)=f(x)g(x). Prove that Lim x->c h(x)=0.
Relevant Equations:
|x-c|<δ -> |f(x)-0|<ϵ
|g(x)|<K
|x-c|<δ -> |f(x)-0|<ϵ
|g(x)|<K
|g(x)||f(x)-0|<K*ϵ
The product of the absolute values equals the absolute value of the product.
|f(x)g(x))|=|h(x)|<K*ϵ
For a range of input values within δ of c, the corresponding outputs can be made within Kϵ of 0. What if g(x) has a jump at x=c but is bounded by K. If approaching x=c from the right g(x) approaches L1 where |L1|<K and if approaching x=c from the left g(x) approaches L2 where |L2|<K. Will the above inequality holds because f(x)=0 and g(x) is bounded though discontinuous fg is again a zero function.