Setting We work on a filtered probability space ( Ω , F , ( F ) t ∈ [ 0 , T ] , P ). Let ( X n ) n and X be finite variation processes such that X n converges pointwise to X, i.e. lim n → ∞ X t n ( ω ) = X t ( ω )for all ( ω , t ) ∈ Ω × [ 0 , T ], and such that ( ‖ X n ‖ ) n converges pointwise to ‖ X ‖ (where ‖ ⋅ ‖ t denotes the total variation on [0,t]).Question Does this imply that sup n ‖ X n ‖ &lt; ∞ pointwise?

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Setting We work on a filtered probability space   (  Ω  ,      F    ,  (      F        )          t      ∈      [      0      ,      T      ]        ,  P  ). Let   (      X    n        )    n   and X be finite variation processes such that       X    n   converges pointwise to X, i.e.      lim          n      →      ∞            X    t    n    (  ω  )  =      X    t    (  ω  )for all   (  ω  ,  t  )  ∈  Ω  ×  [  0  ,  T  ], and such that   (  ‖      X    n    ‖      )    n   converges pointwise to   ‖  X  ‖ (where   ‖  ⋅      ‖    t   denotes the total variation on [0,t]).Question Does this imply that       sup    n    ‖      X    n    ‖  &amp;lt;  ∞ pointwise?

Alisa Gilmore · Accepted Answer

Yes, this follows from the fact that       lim          n      →      ∞        (  ‖      X    n    ‖      )    n    =  ‖  X  ‖  &amp;lt;  ∞ pointwise. This implies there exists   N  =  N  (  ω  ) such that for all   n  ≥  N we have       |    ‖      X    n    (  ω  )  ‖  −  ‖  X  (  ω  )  ‖      |    &amp;lt;  1. Therefore   ‖      X    n    (  ω  )  ‖  &amp;lt;  1  +  ‖  X  (  ω  )  ‖ for all   n  ≥  N, and hence                              sup          n                ‖                  X          n                (        ω        )        ‖                            ≤                  sup                      n            ≤            N                          ‖                  X          n                (        ω        )        ‖        +                  sup                      n            ≥            N                          ‖                  X          n                (        ω        )        ‖                                          ≤                  sup                      n            ≤            N                          ‖                  X          n                (        ω        )        ‖        +        1        +        ‖        X        (        ω        )        ‖        &amp;lt;        ∞        .

Setting We work on a filtered probability space ( <mi mathvariant="normal">Ω

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