so let X(n) be a sequence. Let Y(n) = inf{x(i) : 1 <= i <= n}.(adsbygoogle = window.adsbygoogle || []).push({});

show y(n) is decreasing.

ok, first of all, how do I read this problem?

do I say in words, Y(n) is equal to the infima of x(i), where i is between 1 and n?

so what does that mean?? no idea where to start.

ok, so here is how I usually start. I read each sentence, and try to find out what it is by definition, and then try to connect:

here is attempt:

-ok, so X(n) is a sequence, so we have X(1)...X(2)...X(i)....to X(n).

-and Y(n) is the infima(the lower bound) of X(i), where i could be, and including 1 to n.

-so can I just say Y(n) = inf{X(i)}, where i is arbitrary and between 1 or n.

ok, so I need to show that y(n) is a decreasing...meaning y(1) < y(j) < y(n) right?

ok, so how do I do? I mean, I really have no idea on how else to do this besides defintion by definition, so it takes a while...and I still dont know how to connect....

edit: ok, so here is my attempt at the sollution.

we need to prove that y(n) is decreasing. This actually means that y(1) < y(2) < y(i) < y(n) right?

so by definition, if the inf of x(i) = blah, then y = blah right? so basically, Y(n) is the greatest lower bound of X(n).....

but what does this say about whether Y decrease or increase?

edit again, man, after 1 hour, I give up...but here is the closest thing I can think of. so if Y(n) is the infimum, that means it is the greatest lower bound. So if we let Y(1) Y(2)......Y(i)....Y(n), that obviously means it is decreasing and is going to the infimum....right?

im moving on to the next problem, but can someone clear this up for me?

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# Homework Help: How to prove this?

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