12 Proofs: Gluing and Whitney Machinery

12.1 Theorem 11.23

We begin with Theorem 11.23.

Proof: Using the Lojasiewicz inequality, (see Chapter 8), there exists a Euclidean motion upper A for which we have

StartAbsoluteValue phi left-parenthesis x right-parenthesis minus upper A left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to c Subscript upper K Baseline delta Superscript 1 slash upper C Super Subscript upper K Superscript Baseline comma x element-of upper E period

Without loss of generality, we may replace phi by phi o upper A Superscript negative 1. Hence, we may suppose that

StartAbsoluteValue phi left-parenthesis x right-parenthesis minus x EndAbsoluteValue less-than-or-equal-to upper C prime Subscript upper K Baseline delta Superscript 1 slash upper C Super Subscript k Superscript Baseline comma x element-of upper E period

Now we will employ a similar technique to the proof of Lemma 9.15.

Let theta left-parenthesis y right-parenthesis be a smooth cut off function on double-struck upper R Superscript d such that theta left-parenthesis y right-parenthesis equals 1 for StartAbsoluteValue y EndAbsoluteValue less-than-or-equal-to 1 slash 100, theta left-parenthesis y right-parenthesis equals 0 for and with for all . Then set

The summands are smooth and have pairwise disjoint supports ...

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