The same technique can be used to give an inductive proof of the volume formula.
3.
He gives the first non-inductive proof of the binomial expansion for integer exponent using combinatorial arguments.
4.
This is a great help, as the system has an unproductive tendency to wander down infinite chains of inductive proofs.
5.
This is especially true in inductive proofs, where the given expression is taken to be the inductive hypothesis, and the target expression the inductive conclusion.
6.
A proof such as a deductive proof, an inductive proof, a proof by contradiction, and a proof by construction are the usual stuff that a mathematician follows and accepts readily.
7.
This proof tree can be shown to contain at most 4 \ log _ 2 n-4 values other than 2 by a simple inductive proof ( based on Theorem 2 of Pratt ).
8.
Of course in practice many inductive proofs / recursive constructions are trivial at the limit step, and start with a well-understood object at the base, so you write down only the successor step.
9.
An inductive proof generally shows that some property is true for any " finite " integer " n "; such a proof can't show that it's true if " n " is infinite.
10.
The all-at-once proof " does " have the merit of characterizing the points that are kept, but the inductive proof shows the structure of what has to be removed ( and in particular the structure of countable closed sets in general ).
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