How to make sure I'm not using any proprietary software after installation?

vrms (Virtual Richard M. Stallman) to the rescue:

sudo apt install vrms
vrms

and you'll get something like:

      Non-free packages installed on computer-name

app-shortname1                    Application long name 1
app-shortname2                    Application long name 2
app-shortname3                    Application long name 3

       Contrib packages installed on computer-name

app-contrib1                      Application Contrib Name 1

  3 non-free packages, 0.4% of 2381 installed packages.
  1 contrib packages, 0.1% of 2381 installed packages.

that will ensure you do / do not have any proprietary software installed.

_{Note: CPU firmware patches are non-free but absolutely needed to protect you from some processor flaws and their ilks, so don't go overboard! ;-)}

If you're keen on having only Free Software (although you used the term Open Source) on your computer, you may want to consider installing Ubuntu with the "Free Software Only" option. You can find out more over at this question.