Bounded Function: A function is said to be bounded if it is upper as well as lower bounded.
Fixed point : x is called fixed point of a function f if, f(x)=x i.e. fixed point is an element in domain of f which is mapped to itself.
Since f(x)=x, so to find fixed point x, you need to find solution of f(x)-x=0. Graphically, fixed point is the point of intersection of graphs of y=f(x) and y=x.
Because f is bounded and continuous in [0,∞], you can clearly see that graph of y=f(x) will surely intersect y=x for some x in [0,∞]. Note that f must be bounded as well as continuous to guarantee that it intersects with y=x and hence has a fixed point.