The Scientific Method, Proof, and Skepticism

About a year ago I was having a conversation with a friend who told me that science had proven that God was not necessary for the universe to come into being. He concluded from that God is not required to explain the existence of the universe, and he is justified in his belief that God does not exist. He claims that an honest look at the evidence will lead to this conclusion (implying that other conclusions are not honest evaluations of the scientific data, and that they stifle scientific progress).

On the other hand, about a month ago I was in a conversation with a person who told me that science can't prove anything, and he must be skeptical of everything that scientists say. He believes that he is justified in rejecting many of the commonly accepted-as-true theories in the scientific world in favor of one that the scientific community, as a whole, has rejected. He claims that this is a humble and honest approach to science (implying that all other approaches to science are dishonest, and only skepticism promotes scientific progress).

Validity of the Process of Elimination

I want to take a few minutes to discuss the process of elimination regarding everyday life, science, and philosophy. 

As most of you know, I work in the Information Technology (IT) department at my company. The other day I was doing some troubleshooting for one of our graphic artists. She called me and said that her monitor had started flickering. She stated that she thought that there was something wrong with the monitor and wanted it replaced.  Just to get this on the table now, I was not thrilled with having to replace this specific monitor. It is one of the more expensive ones in the company.

Starting with that thought, I made a list of the possible causes in my head: cables, video card, specific monitor input (it has two), or software on the PC (that could be any range of possibilities). I begin going through some troubleshooting steps to eliminate the possible causes: I reboot the computer; I check (and replace) the cables; I check a different input on the monitor; I try the other output on the video card; check some settings... None of those fixed the problem.