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Proving "for all" statements
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Prove the result for some cases and deduce it for the rest
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Use the continuity method
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[QUICK DESCRIPTION] If you can show that the set of parameters obeying a property $P$ is non-empty, open, and closed, and the parameter space is connected, then $P$ must be obeyed by all choices of the parameter. Thus, for instance, if one wants to prove a property $P(t)$ for all $t$ in some interval $I$, it suffices to establish the following three facts: * A base case $P(t_0)$ for some $t_0 \in I$; * (Openness) If $P(t)$ is true for some $t \in I$, then $P(t')$ is true for all $t' \in I$ sufficiently close to $t$; * (Closedness) If $P(t_n)$ is true for some sequence $t_n\in I$ converging to a limit $t \in I$, then $P(t)$ is also true. [PREREQUISITES] Point-set topology; partial differential equations [EXAMPLE] '''Problem''': (Analytic continuation) Show that a real-analytic function $f(x)$ that vanishes to infinite order at one point $x_0 \in \R$, is identically zero. '''Solution''': Let $P(x)$ denote the assertion that $f$ vanishes to infinite order at $x$. Then, by definition of real analyticity, if $P(x)$ holds for some $x$, then $f$ vanishes within the radius of convergence of the power series at $x$, and so $P(y)$ must then hold for all $y$ in a neighbourhood of $x$. On the other hand, since all the derivatives of $f$ are continuous, if $P(x_n)$ holds for a sequence $x_n$ converging to a limit $x$, then $P(x)$ also holds. Thus the set of $x$ where $P(x)$ is true is non-empty, open, and closed, and is hence all of $\R$. [EXAMPLE] (Solving an ODE in a potential well) [GENERAL DISCUSSION] This method can be viewed as a continuous analogue of [[Induction front page|mathematical induction]] (or conversely, induction can be viewed as a discrete analogue of the continuity method). See also [[A non-trivial circular argument can often be usefully perturbed to a non-circular one]].
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