Convergence of xy Product in $\ell^2$

[tylerc1991]

Homework Statement

Show that if x=(x₁, x₂,...) and y=(y₁, y₂,...) \in \ell² then The sum from i=1 to infinity of |x_iy_i| converges

Homework Equations

The Attempt at a Solution

since x and y are elements of l^2 then The sum from i=1 to infinity of (x_i)² and (y_i)² both converge, which implies that the sum of the product xy converges

Is this right? and more importantly, is it enough? Thank you for your help.

 

[Dick]

Homework Statement

Show that if x=(x₁, x₂,...) and y=(y₁, y₂,...) \in \ell² then The sum from i=1 to infinity of |x_iy_i| converges

Homework Equations

The Attempt at a Solution

since x and y are elements of l^2 then The sum from i=1 to infinity of (x_i)² and (y_i)² both converge, which implies that the sum of the product xy converges

Is this right? and more importantly, is it enough? Thank you for your help.

Restating the problem is hardly a proof. Think about using Cauchy-Schwarz.