The conclusion of the implication is an existentially quantified statement.

To prove the conclusion using the proof by contradiction, we assume that we can put k+1 balls into k boxes such that no box contains 2 or more balls.

The statement to be proved is a universally quantified implication.

The previous statement contradicts the hypothesis that we can put k+1 balls into k boxes.

To prove the implication using the direct proof technique, we assume that we have put k+1 balls into k boxes.

Proof

Since there are k boxes and each box contains at most 1 ball, we can put at most k balls into these k boxes.

The assumption is not true and there must be a box that contains at least 2 balls.

To use the UG rule, we consider an arbitrary positive integer k.

QED