Something necessary is something that exists and cannot possibly-not exist.
Something contingent is something that possibly exists and possibly does not exist. Contingency should not be conflated with "dependency."
Here is the argument:
1. Something presently exists. (Premise)
2. Something cannot come from nothing. (Premise)
3. Either everything that exists is contingent, or else there exists at least one necessary entity N. (Definition)
4. Necessarily, there was never a past time at which nothing existed. (From 1 and 2)
5. Possibly, there was a past time at which nothing contingent existed. (Premise)
6. Therefore, a necessary entity N exists. (From 4 and 5)
Consider this reductio ad absurdum (reduction to the absurd):
7. N does not exist. (Assumption)
8. Possibly, there was a past time at which nothing existed. (From 3, 5, and 7)
9. (8) contradicts (4).
10. Therefore, (7) is false. (From 8 and the law of non-contradiction)