There’s a very simple check you can do to see whether the battery or the charging system (alternator, regulator and wiring) is at fault here. With the engine running, place the probes of a multi-meter (switched to measure voltage) on the relevant battery terminals. If you see a figure of anywhere between 13.2 and 14.7 volts, the charging system is doing its job. You might need to rev the engine slightly off idle to see these numbers, but the voltage going into the battery should be in that range. At this point, the battery becomes the main suspect, and a worn out battery will often neither hold a charge nor accept one.