# Asking for help in proof of part of a Theorem in Apostol Mathematiccal Analysis ( A part which is not proved)

While self studying Lebesgue Integration from Apostol Mathematical Analysis I am struck on this theorem

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I am not able to understand how if theorem is true for {$$s_n$$ - $$s_1$$} then it's true for $$s_n$$ also.

I tried by putting { $$s_n$$ -$$s_1$$} everywhere in theorem where I have {$$s_n$$ } but if $$\int_{I} f$$ = lim n->$$\infty$$ $$\int_{I} ( s_n -s_1)$$ then how does integral of $$s_n$$ also converges to same limit of f ie $$\int_{I} f$$?

I am confused here and can anyone please derive the required justification.

You get that $$\{s_n-s_1\}$$ converges almost everywhere on $$I$$ to a limit function $$f$$ in $$U(I)$$, and $$\int_If=\lim_{n\rightarrow\infty}\int_I(s_n-s_1).$$
Your mistake was to try to prove that $$\{s_n\}$$ converges to $$f$$. But that's not needed. You just have to prove that $$\{s_n\}$$ converges to an element of $$U(I)$$. The obvious candidate is $$f+s_1$$.