I'm having trouble with the following proof, can anyone please help me out?
Let a, b ∈ R with a < b. Suppose that f(x) and g(x) both have a vertical asymptote at x = a. If $\int_{a}^{b} f(x) dx$ is convergent and $\int_{a}^{b} g(x)dx$ is divergent, then prove that $\int_{a}^{b} (f(x)+g(x))dx$ is divergent.
$\int_{a}^{b} f(x) dx = \lim_{A\to a^+}\int_{A}^{b} f(x) dx $ $\int_{a}^{b} g(x)dx = \lim_{A\to a^+}\int_{A}^{b} g(x) dx $
Assume the contrary that it were, then $\lim_{A\rightarrow a^{+}}\displaystyle\int_{A}^{b}(f(x)+g(x))dx$ exists and so is $\lim_{A\rightarrow a^{+}}\displaystyle\int_{A}^{b}(f(x)+g(x)-f(x))$ by a standard limit rule, then we get the convergence of $\lim_{A\rightarrow a^{+}}\displaystyle\int_{A}^{b}g(x)dx$, a contradiction.