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I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):

In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $

I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.

Thanks!

  • 9
    Divide the sum into two parts; "small indices" and "large indices". For large indices use $|x_i-x|<\epsilon$. For small use $|x_i|\le M$.2012-06-08
  • 0
    Check [Cesaro mean theorem](http://en.wikipedia.org/wiki/Ces%C3%A0ro_mean).2013-06-04
  • 0
    https://math.stackexchange.com/questions/207910/prove-convergence-of-the-sequence-z-1z-2-cdots-z-n-n-of-cesaro-means2018-01-01

3 Answers 3