please help wiht math question?????

Posted on 2014-11-27 07:53:00 in Math by MINH

For a real number a and non-empty subset of reals B, define: a + B = { a + b : b is in B }. Show that if B is bounded above, then sup( a + B ) = a + sup B

What I have so far:
since B is bounded, ??B, b ? m, for some real number m

?b?B, a + b ? a + m.

Since a + b is an arbitrary element in a + B, a + B is bounded above by a + m.

Since R has the least upper bound properties, both a + B and B has a supremum.
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The part I'm currently having trouble with is sup(a + B) = sup(B)

My attempt so far is,
?b?B, b ? sup(B) by property of supremum.

?b?B, a + b ? a + sup(B)

hence, sup(a+B) ? a + sup(B)

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