I recently conducted an experiment where I electrolyzed a solution of sodium chloride. I was using a copper coin as the anode, and I observed a pale blue precipitate in the cup when I was done with the experiment. I then deduced the following:
The chloride ions get oxidized at the anode:
$$\ce{2Cl- -> Cl2 + 2e-}$$
The chlorine gas then reacts with the copper anode to form copper(II) chloride which goes into solution as it is completely soluble.
At the cathode, the hydrogen in the water gets reduced to form hydrogen gas: $$\ce{2H2O + 2e- -> 2H2 + 2OH-}$$
I know the above half reactions exist from the list of standard reduction potentials.
Next, the copper(II) ions react with the hydroxide ions being formed to produce copper(II) hydroxide, which is both insoluble and pale blue.
However, this source mentions that the hydronium ions in the water get reduced at the cathode:
$$\ce{2H+ + 2e- -> H2}$$
Is this correct? In my understanding, the concentration of hydronium ions ($\pu{1e-7 M}$) would be much too low, and where would the hydroxide ions come from to form copper(II) hydroxide?