Answered: Show that the nondegenerate interval…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Show that the nondegenerate interval [2, 5] is uncountable.

Expert Solution

Step 1: uncountability of a set

Here first we have to show that $left square bracket 0 comma 1 right square bracket$ is not countable.

If possible let, $A equals open square brackets 0 comma 1 close square brackets space$ is countable ,so A can be expressed as $A equals open curly brackets x subscript 1 x subscript 2 comma x subscript 3 comma........ close curly brackets$

Obviosly $0 less or equal than x subscript i less or equal than 1 comma for all i$

Obseved that $1 space a n d space 1 half space c a n space b e space e x p r e s s e d space i n space t w o space w a y s space colon$ $open curly brackets table row cell 1 equals 1.0000 end cell row cell 1 half equals 0.5000 end cell end table close space a n d space open curly brackets table row cell 1 equals 0.99999... equals 0.9 end cell row cell 1 half equals 0.4999999... equals 0.49 end cell end table close$

Now write decimal expressionns of each $x subscript i space i n space t h e space f o l l o w i n g space f o r m space colon$

$x subscript i equals 0. x subscript i 1 end subscript x subscript i 2 end subscript x subscript i 3 end subscript x subscript i 4 end subscript...... x subscript i m end subscript space...... comma w h e r e space x subscript i j space end subscript element of open curly brackets 0 comma 1 comma 2 comma.. comma 9 close curly brackets$