Hello everyone,(adsbygoogle = window.adsbygoogle || []).push({});

is the following an equivalent definition of the supremum of a set M, M subset of R?

y=sup{M} if and only if

given that y is an upper bound of M and x is any real number,

y >= x implies there exists m in M so that m >=x.

pf:

Let x_n be a sequence approaching y from the right. Then

for each x_n, there exists m_n in M so that m_n >=x_n.

Since y is an upper bound of M, then we have that y= lim m_n >= lim x_n.

Therefore, if m' is any another upper bound, then m'>=y for all m in M.

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Equivalent definition of the supremum

**Physics Forums | Science Articles, Homework Help, Discussion**