i have searched in many books but i did not find a proof for the statement in the title. I know its linked with Cauchy's theorem, but i need a full and reasoned proof. Thanks.
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Welcome to MSE. What have you tried, and where did you fail? – jvdhooft Jun 02 '17 at 08:37
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i have tried to use the method of majorants to solve the problem but failed. – Dor Gal Swissa Jun 02 '17 at 08:45
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The Picard iteration in the proof of the Picard-Lindelöf theorem gives you a sequence $(\varphi_i)_{i\geq 0}$ converging in the sup-norm to $x$. However, all the $\varphi_i$ are analytic (prove this by induction, using that $f$ is analytic) and therefore $x$ is analytic as well.
Added much latter: A more complete answer about the real-analytic Picard-Lindelöf theorem can be found here Why does rational dependence of $f'$ on $f$ imply that $f$ is real-analytic?
Severin Schraven
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@MatheusManzatto It is a standard fact, that analyticity is preserved under the supremum norm (try to prove this using the Morera criterion, you can look here https://math.stackexchange.com/questions/368664/uniform-limit-of-holomorphic-functions ) – Severin Schraven Jul 22 '17 at 22:04
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@MatheusManzatto Analyticity is a local property, i.e. we can check it on a small neighborhood. In fact, the Picard iteration only works in a restricted neighborhood. Please let me know if I have to add more details to make it clear – Severin Schraven Jul 22 '17 at 22:38