Answer to Question #210242 in Real Analysis for Nikhil

Question

Suppose that the sequence (sn) converges to s and sn ≤ A for every n. Show that s≤A

Answer 1

Given sequence (s_n) is convergent and bounded by A

i.e. s_n $\leq$ A for all n

(s_n) is convergent therefore by definition -

($\lim_{n \to \infty}$ s_n = L ,if for every number  $\epsilon$ > 0.there is an integer N such that $\mid$ s_n - L $\mid$ < $\epsilon$ whenever n > N)

since the sequence is bounded hence every term is less than A

and by definition of limit after n > N, $\mid$ s_n - L $\mid$ < $\epsilon$ (i.e. the difference between s_n and L is negligible OR s_n is almost equal to L)

therefore,

$\mid$ L $\mid$ $\leq$ A