Orthogonal problem

Problem: Suppose the sequence {Uk} in R^n converges to u. Let v be in R^n that is orthogonal to each Uk. Prove that v is also orthogonal to u.

My proof:

Now, we want that Uk -> to u, so ||Uk - u|| -> 0, and <Uk,v> = ||Uk|| ||v|| = 0. And I need <v,u> = 0.

Now <v,u> <= ||v|| ||u||. Okay, I'm lost now. I have all the pieces, but I can't seem to put them together well...