show that the sum of two increasing functions is increasing
if f is increasing, f(x+h) > f(x) for h>0
so, if f and g are both increasing
f(x+h) + g(x+h) > f(x) + g(x)
To show that the sum of two increasing functions is increasing, we need to prove that for any two increasing functions f(x) and g(x), their sum h(x) = f(x) + g(x) is also increasing.
To do this, we can use the definition of an increasing function. A function f(x) is said to be increasing if for any two values a and b in its domain such that a < b, f(a) <= f(b).
Now, let's consider two values, a and b, in the domain of h(x) such that a < b. We want to show that h(a) <= h(b).
Using the definition of h(x) = f(x) + g(x), we have:
h(a) = (a)
h(b) = f(b) + g(b)
Since f(x) and g(x) are increasing functions, we know that f(a) <= f(b) and g(a) <= g(b) because a < b. Hence, we can write:
(a) <= f(b) + g(b)
Therefore, we have h(a) <= h(b), showing that h(x) = f(x) + g(x) is also an increasing function.
Note: It's important to keep in mind that this proof assumes that the functions f(x) and g(x) are defined on the same domain, and that their sum is also well-defined within that domain. Additionally, it's important to ensure that the functions f(x) and g(x) are increasing over their entire domain and not just for specific intervals.