Why the sequence $(E_n)$ exists

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Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $(X,\|.\|)$ be a reflexive Banach space.

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I did not understand why the sequence $(E_n)$ exists.

An idea please.

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Simply take $$ E_n =\{x : \|f(x)\| + \|g(x)\| \leq n \}. $$