# Let S be as set of real numbers , and let \${x_n}\$ be a sequence which converges to l

Let S be as set of real numbers , and let \${x_n}\$ be a sequence which converges to l. Suppose that for every \$n inmathbb{N},x_n\$ is an upper bound for S . prove l is an upper bound of S

And also Suppose \${x_n}\$ is a sequence in S such that \$x_n to 1\$ and 1 is an upper bound of S. Show that \$1 = text{lub}(S)\$

my idea:

since {x_n} is converges then \$|x_n-l|<epsilon,\$ for each \$n ge N\$

Also \$x_n \$ is upper bound of S then \$x<x_n, forall nin N\$

but how to we processed from here