How to find l i m n → ∞ s u p x &gt; 0 1 ( 0 , t n ) ( x ). It is given that t n ≥ 0 , t n → 0 as n → ∞ and 1 denotes indicator function. It seems that limit should be simply zero but how to justify it? I'm facing this problem while trying to prove a result in banach spaces.

Question

How to find   l  i      m          n      →      ∞           s  u      p          x      &amp;gt;      0            1          (      0      ,              t        n            )        (  x  ). It is given that       t    n    ≥  0  ,      t    n    →  0 as   n  →  ∞ and 1 denotes indicator function. It seems that limit should be simply zero but how to justify it? I&#039;m facing this problem while trying to prove a result in banach spaces.

Ethen Valentine · Accepted Answer

For any   a  &amp;gt;  0,      sup          x      &amp;gt;      0                  1              (      0      ,      a      )        (  x  )  =  1.Hence,      lim          n      →      ∞            sup          x      &amp;gt;      0                  1              (      0      ,              t        n            )        (  x  )  =      lim          n      →      ∞        1  =  1not 0.On the other hand,      sup          x      &amp;gt;      0            lim          n      →      ∞                  1              (      0      ,              t        n            )        (  x  )  =      sup          x      &amp;gt;      0        0  =  0which is maybe what you meant?

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