Find the mistakes in the proof fragments. Theorem: For any integer n≥0,

1+2+2^{2}+⋯+2^{n}=2^{n+1}−1. “Proof (by mathematical induction):

Let the property P(n) be 1+2+2^{2}+⋯+2^{n}=2^{n+1}−1.

Show that P(0) is true: The left-hand side of P(0) is 1+2+2^{2}+⋯+2^{0}=1 and the right-hand side is 2^{0+1}−1=2−1=1 also. So P(0) is true."