Is My Proof for Convergence to the Mean Valid?

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SiddharthM
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Convergence in the Mean. I attached a picture of a handwritten proof because I don't know how to use Latex and there is too much hullabaloo to type it out without the notation.

I'm wondering if my proof is legit - i make the size of 1/n be less than epsilon/kM...i'm not sure...hmmmmmmm, I think it's fine though because that k is fixed.

Let me know if my proof is satisfactory.

a .jpg file is attached with the proof.
 

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so it's correct?
 
can someone please take a serious look at this and tell me whether or not the proof is fine?