The Proofs of Triangle Inequality Using Binomial Inequalities

In this paper, we introduce the different ways of proving the triangle inequality ku − vk ≤ kuk + kvk, in the Hilbert space. Thus, we prove this triangle inequality through the binomial inequality and also, prove it through the Euclidean norm. The first generalized procedure for proving the triangle inequality is feasible for any even positive integer n. The second alternative proof of the triangle inequality establishes the Euclidean norm of any two vectors in the Hilbert space.