confusion about sequences bounds

[random grove]

(Un) is said to be bounded if there exists M such that the absolute value of Un <= M , meaning Un is between minus M and M .. why does Un has to be bounded with the same value , for example if Un is between 2 and 3 Un is indeed bounded above and below but abso value of Un isnt <= 3

[void rivet]

If U_n ist between 2 and 3 for all N. Then |U_n| = U_n =< 3

[random grove]

what if its between -4 and 3

[void rivet]

Then you can simply take M= max(|lower bound|, |upper bound|)

[random grove]

so M doesn't stand for upper and lower bound ?

if yes then, we have no problem here

yeah my question was if M is the bound but turns out no

but still there's sup Un and inf Un ?

they aren't equal to M

[void rivet]

No, M is just an upper bound for |U_n|. That means -M <= U_n <= M. But you are right that sup U_n <= M and inf U_n >= -M, so they don't have to be M

[random grove]

btw , if they ask me to make sure if the sequence converges to l

should i prove it using the definition of the limit ?

absolute value of un - l < epsilon

[void rivet]

Depends on the sequence?

[random grove]

humm

[void rivet]

Do you have a specific example?

[random grove]

so for example if un = 2n+1/n+3

do i have the right to consider Un as f(n)

and calculate the limit of f(n)

and say that lim Un = lim f(n)

[void rivet]

Do you mean (2n+1)/(n+3)?

[random grove]

yes

[random grove]

or do i have to use the definition and epsilon ?

[void rivet]

You probably just started Calc 1?

[random grove]

yes 😭

[void rivet]

Tbh I can't say what you are allowed to use or not. At the beginning you are very restricted in the methods. So you know better than I do what is allowed

[random grove]

ok so is this method correct ?

in general math

[void rivet]

For this one I would devide numerator and denominator by n and show separately that the limits exist. For this you can use that 1/n -> 0 because you have already shown it