# Limit of a sequence

1. Feb 7, 2012

### cragar

1. The problem statement, all variables and given/known data
Let [x] be the greatest integer ≤x. For example $[\pi ]=3$
and [3]=3
Find $lim a_n$ and prove it.
a) $a_n=[\frac{1}{n}]$
b) $a_n=[\frac{10+n}{2n}]$
3. The attempt at a solution
for the first one it will converge to zero.
so can I write $\frac{1}{n}< \epsilon$
then I can just pick an n large enough to make that work.
for part b, it also looks like it will converge to zero, but a little slower.
so $\frac{10+n}{2n}< \epsilon$ and then solve for n in terms of ε.
Or is there something I am not taking into account with the greatest integer deal?

2. Feb 7, 2012

### Deveno

for the first one:

you are NOT trying to "solve for ε in terms of n", but rather, the other way around. you need to find a (possibly large) value for N so that n > N makes 1/n "small" (less than epsilon). epsilon is arbitrary, but assumed as given.

for the second one, i would note that:

(n+10)/2n = 1/2 + 5/n.

if you find an N such that n > N means 5/n < 1/2,

wouldn't [(n+10)/2n] = 0 for all such n?

it seems to me the floor function "speeds up" the convergence, instead of making it slower.

3. Feb 7, 2012

### cragar

thanks for your help, ok
so on the second one $.5+\frac{5}{n} < \epsilon$
so then I just pick an n large enough to make it less than epsilon.

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