Multiply infinity by a positive number

  • #1

Homework Statement

Prove that if limXn = +∞ and limYn>0 then limXnYn=+∞

The Attempt at a Solution

limXnYn = limXnlimYn = (c)(+∞) where c is a positive real number

I know in my head that a positive number multiplied by infinity is positive, but I am unsure how to prove this and we have not yet done this particular example in class.
  • #2
instead of thinking of ∞ as a number, think of:

[tex]\lim_{n \to \infty} x_n = \infty[/tex]

meaning, no matter how large a positive real number N we choose, for all large enough n, xn > N.

now, suppose

[tex]\lim_{n \to \infty} y_n = L > 0[/tex]

for large enough n, we can ensure that yn > L/2 > 0.

can we make xnyn larger than any positive real number M?

(what happens if we pick n so that xnis larger than 2M/L, and yn is larger than _____?)

Suggested for: Multiply infinity by a positive number