1. **State the problem:** We need to sort the list $[5, 1, 3, 6, 4]$ using the bubble sort algorithm. 2. **Explain bubble sort:** Bubble sort repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order. This process is repeated until the list is sorted. 3. **Step 1:** Compare 5 and 1. Since $5 > 1$, swap them. List becomes $[1, 5, 3, 6, 4]$. 4. **Step 2:** Compare 5 and 3. Since $5 > 3$, swap them. List becomes $[1, 3, 5, 6, 4]$. 5. **Step 3:** Compare 5 and 6. Since $5 < 6$, no swap. List remains $[1, 3, 5, 6, 4]$. 6. **Step 4:** Compare 6 and 4. Since $6 > 4$, swap them. List becomes $[1, 3, 5, 4, 6]$. 7. **End of first pass:** The largest element 6 is now at the end. 8. **Step 5:** Compare 1 and 3. Since $1 < 3$, no swap. List remains $[1, 3, 5, 4, 6]$. 9. **Step 6:** Compare 3 and 5. Since $3 < 5$, no swap. List remains $[1, 3, 5, 4, 6]$. 10. **Step 7:** Compare 5 and 4. Since $5 > 4$, swap them. List becomes $[1, 3, 4, 5, 6]$. 11. **End of second pass:** The second largest element 5 is now in place. 12. **Step 8:** Compare 1 and 3. Since $1 < 3$, no swap. List remains $[1, 3, 4, 5, 6]$. 13. **Step 9:** Compare 3 and 4. Since $3 < 4$, no swap. List remains $[1, 3, 4, 5, 6]$. 14. **End of third pass:** No swaps needed, list is sorted. **Final sorted list:** $[1, 3, 4, 5, 6]$. Bubble sort has successfully sorted the list in ascending order by repeatedly swapping adjacent elements that are out of order.