Having trouble understanding the iron-carbon phase diagram At room temperature, alpha and alpha+$\ce{Fe3C}$ are present (depending on carbon content), which is ferrite and pearlite. In the lab we polished and etched samples of carbon steel to observe the microstructure, and saw both ferrite grains AND pearlite grains. Not unlike the following:



However, from my understanding of the phase diagram, only one phase should be present. For any steel with less than 0.02% carbon we should see only ferrite, and for steel with >0.02% carbon we should see only pearlite. Though this was not the case, and rather there was a mix of pearlite and ferrite, with the higher carbon content steel having a higher proportion of pearlite.

It would be much appreciated if someone could explain to me what is happening here. Links to further reading would be helpful but not required.
For any steel with less than 0.02% carbon we should see only ferrite, and for steel with >0.02% carbon we should see only pearlite.


You have got a small misunderstanding here. The maximum solubility of carbon at $723\ \mathrm{^\circ C}$ is 0.02%. At room temperature it is very low ($0.005%$ at $0\ \mathrm{^\circ C}$). Also for carbon level between 0.02 and 0.8 you wont see only pearlite, you will see a combination of ferrite and pearlite for sure. Only at 0.8% you can observe pure pearlite(100%). Also note that the phase diagram is in such a way that if you cool from the ferrite+ austenite region at $800\ \mathrm{^\circ C}$ with carbon content &lt; 0.02% we can still get a ferrite + pearlite + even austenite/martensite.
Phase diagrams give an ideal picture. In real situations, it is very tough to get very slow cooling rates to get the equilibrium structure.