Convergence of orthagonal vectors

[cap.r]

Homework Statement

{u_k} is in Rⁿ and converges to u in Rⁿ

let v be in Rⁿ and v is orthogonal to each u_k.

prove v is orthogonal to u

Homework Equations

just definition of convergence. and orthogonality. <v,u>=0 if v is orthogonal to u.

The Attempt at a Solution

u_k converges so it is cauchy, so it's terms are getting closer to each other.

for epsilon>0 , there exists k>= k₀ st. ||u_k-u|| < epsilon
so if v is orthogonal to u_k then u is orthogonal to each term in u_k. but the terms of u_k are getting closer to u. so if v is orthogonal to a u_k that is very close to u, then it is also orthogonal to u.

this proof is in no way formal but i think i have the right idea. can some one please help rewrite this?

 

[aPhilosopher]

It sounds like you want to argue that if u_k converges to u, then <u_k, v> converges to <u, v>. Does that sound right?

if so, let h be an arbitrarily small vector. what happens to <u + h, v> = <u, v> + <h, v> as h goes to zero? The magnitude can vary of course, But you just want an upper and lower bound on it anyways. Each magnitude of h represents a neighborhood of u that contains every element of u_k for k sufficiently large. (this of course comes from the definition of a limit)

Alternatively, you might be able to do something cool with the fact that <u - u_k, v> = <u, v>. That that is useful is just a guess on my part though.