Question on a technical lemma: Operator with bounded inverse

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I'm studying about Hilbert spaces this semester and now we're talking about "Resolvent and Spectrum". Here is a Lemma:

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I understand the whole proof except for the part : "$\;T^{n+1} \to 0\;$ in the operator norm". Why does this hold?

I'm sure I'm missing something here but I can't figure out what. Any help would be valuable. Thanks in advance!!!