Is my car battery almost dead?

I'm not sure how long I've had it. 2 weeks ago, I brought the car to the mechanic to change tires. He also cleaned the corrosion foo the battery (now there is that red substance on the terminal, I'm guessing it's to keep them from getting corroded again). Anyway, since they cleaned remove the corrosion (I'm not sure how much was on it), it's the third time that I cannot start it (the last 2 times, I just had to turn the key on off position, then turn it back on, and it was ok), but now, the car doesn't want to start at all, nothing works except for the radio works off and on and the digital mileage on the dashboard.