Take B a compact set large enough that |
Ti(B) ⊆ B for i = 1, ..., N. |
This is always possible because for large enough B, the effect of translation will be negligible compared with contraction. |
Then |
B ⊇ T1(B) ∪ ... ∪ TN(B) = T (B) |
Applying T to this inclusion gives |
T (B) ⊇ T (T (B)) = T 2(B) |
Continuing to apply T and combining the inclusions gives |
B ⊇ T (B) ⊇ T 2(B) ⊇ T 3(B) ... |
and so |
B, T (B), T 2(B), T 3(B), ... |
is a nested sequence of nonempty compact sets. Then |
A = ∩T k(B) |
is a nonempty, compact set. |
Return to the proof of the theorem.