Inequality between two sequences preserved in the limit? [duplicate]

real-analysis sequences-and-series limits inequality

Yes. If you assume that $a < b$, you can take $\varepsilon = (b-a)/2$ and use the definition of the limit of a sequence to come up with a contradiction, i.e. find an $n$ such that $a_n < b_n$.

Related

The image of the diagonal map in scheme
Prove the third isomorphism theorem
Simple $M_n(D)$-module with $D$ a division ring
A commutative ring whose all proper ideals are prime is a field. [closed]
Complex ($\mathbb C$) least squares derivation
Size of the product of two subgroups
Polar Coordinate Transformation - Motivation
Expected days to finish a box of cookies
Is $\cos(x) \geq 1 - \frac{x^2}{2}$ for all $x$ in $R$?
Injective Cogenerators in the Category of Modules over a Noetherian Ring
Quotient of a Banach space $X$ gets quotient topology under standard norm induced from $X$.
Which meromorphic functions are logarithmic derivatives of other meromorphic functions?