- #1

synkk

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my attempt thus far

given ## \epsilon > 0 ## ##\exists N \in \mathbb{R} ## s.t. n > N implies ## |x_n - l | < \epsilon ## where l is the limit of ## x_n##.

now we need to prove that for given ## \epsilon > 0 ## ##\exists N_1 \in \mathbb{R} ## s.t. ##n > N_1 ## implies ## |x_{n+1} - m | < \epsilon ## where m is the limit of ## x_{n+1} ## and m = l

heres what I have thus far

consider ## |x_n - l |## :

## |x_n - l | = |x_n - l + x_{n+1} - x_{n+1} | = |(x_n - x_{n+1}) + (x_{n+1} - l ) | \leq |x_n - x_{n+1} | + |x_{n+1} - l| \leq |x - x_{n+1} | + |x_{n+1} - l | \leq |x_{n+1} - x | + |x_{n+1} -l | ##

any ideas how to proceed? or if I'm even doing it correctly