Equivalent definition of the supremum

[submartingale]

Hello everyone,

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

 

[micromass]

This is not true. Specifically, if y=sup(M), then it does not need to holds that y>=x implies m>=x for an m.

Indeed, take y=x.

 

[submartingale]

This is not true. Specifically, if y=sup(M), then it does not need to holds that y>=x implies m>=x for an m.

Indeed, take y=x.

If you take y=x, then there exists m in M so that m>=x=y. But y is an upper bound of M, so y=x=m.

 

[micromass]

Take A=]0,1[, then y=1 is a supremum. Does there exist an m in A such that m>=y??

 

[submartingale]

Take A=]0,1[, then y=1 is a supremum. Does there exist an m in A such that m>=y??

What if we replace it by

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.

Thanks

 

[micromass]

What if we replace it by

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.

Thanks

That's indeed correct.