Various Methods for Proving a Sequence of Real Numbers is Convergent

Except for 3, all of these methods require that we already know the value of the limit, and we then verify that our suspicion is correct.

We can use the following definitions/theorems the conditions matches:

1. 3.1.3, 3.1.5. (Often difficult)
2. 3.1.10: Obtain by a multiple of the terms in a sequence converging to $0$
3. 3.1.9, 3.2.3, 3.2.9, 3.2.10: Obtain from other converging sequences.
4. 3.2.7: Squeeze between two sequences that has the same limit.
5. 3.2.11: Ratio test.
6. 3.3.2: Sequence is monotone.
7. 3.4.9: All convergent subsequence converges to the same limit.
8. 3.4.2: Subsequence from a sequence converges to the same limit.
9. 3.5.5, 3.5.8: Cauchy & contractive sequence

Examples

• Using 3.1.3:
  • 00004.excalidraw
  • 00005.excalidraw
  • 00003.excalidraw
  • Pasted image 20250310083002
• Using 3.1.10:
  • 00002.excalidraw
  • 00001.excalidraw
• Using 3.2.3:
  • 00000.excalidraw
• Using 3.2.10:
  • 00006.excalidraw
• Using 3.3.2:
  • 00007.excalidraw
  • 00009.excalidraw (recursively defined sequence)
• Using 3.5.8:
  • 3^n). Contractive → convergent