Nature of the improper integral $\int_{0}^{\infty}{\dfrac{1}{t}}dt$

Question

I want to show that $\int_{0}^{\infty}{\dfrac{1}{t}}dt$ is divergent. Now we have a problem of boundedness of $f(t)=\dfrac{1}{t}$ on $t=0$ and we have a problem of boundedness of the domain $(0,\infty)$ So we treat each problem apart by writing $$\int_{0}^{\infty}{\dfrac{1}{t}}dt=\int_{0}^{1}{\dfrac{1}{t}}dt+\int_{1}^{\infty}{\dfrac{1}{t}}dt$$ Now the two parts are Riemann integrals such that $\int_{0}^{1}{\dfrac{1}{t}}dt$ is divergent and $\int_{1}^{\infty}{\dfrac{1}{t}}dt$ is also divergent. The sum of two divergent integrals may or may not converge, so how do we treat this problem. Thank you for your help!!