If x n , x , y are vectors in R m and x n → x, prove that ⟨ x n , y ⟩ → ⟨ x , y ⟩I am assuming one can simply say that x n → x so we get the result but I don’t understand how vectors work and an explanation of how this proof works would go a long way.

Question

If       x    n    ,  x  ,  y are vectors in       R    m   and       x    n    →  x, prove that   ⟨      x    n    ,  y  ⟩  →  ⟨  x  ,  y  ⟩I am assuming one can simply say that       x    n     → x so we get the result but I don’t understand how vectors work and an explanation of how this proof works would go a long way.

Joe Sheppard · Accepted Answer

By Cauchy-Schwarz inequality and   ‖      x    n    −  x  ‖  →  0 as   n  →  ∞                              |                ⟨                  x          n                ,        y        ⟩        −        ⟨        x        ,        y        ⟩                  |                =                  |                ⟨                  x          n                −        x        ,        y        ⟩                  |                ≤        ‖                  x          n                −        x        ‖        ‖        y        ‖        →        0            as   n  →  ∞

If x_n,x,y are vectors in R^m and x_n to x, prove that ⟨x_n,y⟩ to ⟨x,y⟩.

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