Please help 1 Prove by induction that for every n123 1xn 1n

Please help!!!

1. Prove by induction that for every n=1,2,3,.... (1+x)^n >= 1+nx for any x>0

2. Show that the defintion of \"dense\" could be given as

A set E of real numbers is said to be dense if every interval (a,b) contains infinitely many points of E.

Solution

1. Let the given statement is p(n) : ( 1 + x )ⁿ >= 1 + nx --------(1)

this can be proved by mathematical induction in 2 steps

step 1 : basic step Put n=1 in p(n) : p(1) ; 1 + x = 1+x whichis true .Hence p(1) is true ----- I

step 2 : induction step : let p(k) istrue ie (1+x)^k  >= 1+kx

consider LHS of p(k+1) : (1+x)^k+1  = (1+x)^k 1+x) >= (1+kx) (1+x)

.>= 1+ kx +x + kx² >= 1 + (k+1) x

>= 1+ (k+1)x which is the RHS oft he given statement

Hence Induction step is valid -------- II

from I and II the statement P (n) is valid for all n by Mathematical induction

p(n) : (1 +x) ⁿ >= 1 +nx is true for all n