I used half my Sunday for trying to proof the following, but I couldn't find the answer. Can you help me?
Let f(x)=∑∞k=0ak(x−x0)k a power series with radius of convergence ≥0. Show: f(xn)=0 for a sequence of points {xn} with xn→x0,xn≠x0⟹ak=0 ∀k∈N.
Hint: Define for j∈N
f(j)(x):=∞∑k=0aj+k(x−x0)kand use induction to prove for all j∈N:
f(jn)(xn)=0 for all n∈N and aj=0.
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