That's a proof by induction.

Basically, you show it's true for the base step. You've said it is...

i.e. P(1) is true.

Then you assume it's true for an arbitrary value, in this case j...

i.e. P(j) is true.

By showing the next value is also true, you have completed the proof. This is called the inductive step.

Since you said that k = j+1 and P(k) is true, the inductive step is complete and thus the proof is complete.