Proof
Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990), from which the proof below is taken.
Note that it suffices to prove the result for a small interval I = (−ε,ε), since if ψ(t) is a smooth bump function with compact support in (−ε,ε) equal identically to 1 near 0, then ψ(t) ⋅ F(t, x) gives a solution on R × U. Similarly using a smooth partition of unity on Rn subordinate to a covering by open balls with centres at δ⋅Zn, it can be assumed that all the fm have compact support in some fixed closed ball C. For each m, let
where εm is chosen sufficiently small that
for |α| < j. These estimates imply that each sum
is uniformly convergent and hence that
is a smooth function with
By construction
Note: Exactly the same construction can be applied, without the auxiliary space U, to produce a smooth function on the interval I for which the derivatives at 0 form an arbitrary sequence.
Read more about this topic: Borel's Lemma
Famous quotes containing the word proof:
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
—Susan B. Anthony (1820–1906)