Equivalence of Definitions of Norm of Linear Transformation

Theorem
Let $H, K$ be Hilbert spaces.

Let $A: H \to K$ be a bounded linear transformation.

Proof
Let:

From Operator Norm is Finite:
 * $\lambda_4 < \infty$

We will show that:
 * $\lambda_4 \ge \lambda_2 \ge \lambda_1 \ge \lambda_3 \ge \lambda_4$

Inequality: $\lambda_3 \ge \lambda_4$
It follows that the definitions are all equivalent.