# How do I find the greatest lower bound for the sequence #A={\frac{1}{n+10}}_{n=1}^{\infty}#?

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I know the infimum is zero, and I know I need to find #a_{n}(\epsilon)\in A# such that #\forall \epsilon>0:0+\epsilon>a_{n}(\epsilon)# . How do I go about finding #a_{n}(\epsilon)# ?

I know the infimum is zero, and I know I need to find

##### 1 Answer

May 2, 2018

supremum

#=1/11# and infimum#= 0 #

#### Explanation:

We seek upper and lower bounds for the sequence:

# { 1/(n+10) } # for# n in [1,oo)#

If we consider the function:

# f(x) = x+10 #

Then trivially,

Thus we have an upper bound when

supremum

#=1/f(1)=1/11#

And a lower bound as

infimum

#=lim_(n rarr oo) 1/f(n) = lim_(n rarr oo) 1/(n+10) = 0 #