document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019873>Question 1019873</A>: <I><SPAN STYLE=\"word-break: break-all;\"> By giving a proof or a counterexample, determine the truth value of the following<BR>\n" );
document.write( "statement.<BR>\n" );
document.write( "For every pair of natural numbers x and y such that x > 2y there is a natural number z such that<BR>\n" );
document.write( "x > z > y.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #635856 by </B><A HREF=/tutors/aboutme.mpl?userid=LinnW><B>LinnW(1048)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=LinnW><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=LinnW><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=635856\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose x and y are a pair of natural numbers such that x > 2y <BR>\n" );
document.write( "Since y is a natural number, 2y is a natural number<BR>\n" );
document.write( "x must be ( 2y + 1 ) or larger to satisfy x > 2y since 2y + 1<BR>\n" );
document.write( "is the next larger natural number above 2y.<BR>\n" );
document.write( "Let x' = 2y + 1<BR>\n" );
document.write( "Let z = 2y<BR>\n" );
document.write( "We know y < 2y and 2y < 2y + 1<BR>\n" );
document.write( "so y < 2y < 2y + 1<BR>\n" );
document.write( "y < z < x'<BR>\n" );
document.write( "We know x' < x<BR>\n" );
document.write( "so y < z < x or x > z > y </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );